# Dynamic V2X Autonomous Perception from Road-to-Vehicle Vision

Jiayao Tan1, Fan Lyu2, Linyan Li3, Fuyuan Hu1, Tingliang Feng4, Fenglei Xu1,
Rui Yao Corresponding author.

###### Abstract

Vehicle-to-everything (V2X) perception is an innovative technology that
enhances vehicle perception accuracy, thereby elevating the security and
reliability of autonomous systems. However, existing V2X perception methods
focus on static scenes from mainly vehicle-based vision, which is constrained
by sensor capabilities and communication loads. To adapt V2X perception models
to dynamic scenes, we propose to build V2X perception from road-to-vehicle
vision and present _Adaptive Road-to-Vehicle Perception_ (AR2VP) method. In
AR2VP, we leverage roadside units to offer stable, wide-range sensing
capabilities and serve as communication hubs. AR2VP is devised to tackle both
intra-scene and inter-scene changes. For the former, we construct a dynamic
perception representing module, which efficiently integrates vehicle
perceptions, enabling vehicles to capture a more comprehensive range of
dynamic factors within the scene. Moreover, we introduce a road-to-vehicle
perception compensating module, aimed at preserving the maximized roadside
unit perception information in the presence of intra-scene changes. For inter-
scene changes, we implement an experience replay mechanism leveraging the
roadside unit’s storage capacity to retain a subset of historical scene data,
maintaining model robustness in response to inter-scene shifts. We conduct
perception experiment on 3D object detection and segmentation, and the results
show that AR2VP excels in both performance-bandwidth trade-offs and
adaptability within dynamic environments. Our code is available at:
https://github.com/tjy1423317192/AP2VP

### Introduction

The Vehicle-to-everything (V2X) technique (Y. Li and Wang 2022; M.Muhammad and
G.A.Safdar 2018; M.Hasan and H.Lu 2018), facilitating collaboration between
vehicles and various other entities (Y.Li and et al 2021; Y.Yuan and M.Sester
2021), is introduced as a popular means to enhance the perception system for
intelligent driving (A.Geiger 2012; Z.Jiarui and et al 2023). _However,
existing V2X research predominantly centers around static data (barely no
entity or scene change), which inadequately addresses the safety prerequisites
of vehicles in dynamic traffic environments._ To elaborate, a dynamic traffic
environment encompasses two key facets: (1) Intra-scene variations: This
involve the variations within a scene, containing factors like pedestrians in
motion and vehicles executing turns. (2) Inter-scene variations: This pertains
to substantial changes in architecture and lane configurations between
different scenes. An autonomous system should possess the capability to
facilitate driving in dynamic environments, where inaccurate perception may
potentially lead to traffic incidents.

Figure 1: Dynamic Scene Advantage of RSU. RSU demonstrates stability in
perception and possess geographical advantages in inter-scene changes, as
opposed to the continual mobility of vehicles in intra-scene dynamics. With
storage and communication capabilities, RSU stores old scenes data and model,
enabling adaptation to inter-scene changes.

The primary limitation of existing V2X studies in accommodating dynamic
environments stems from their heavy dependence on vehicle-based vision,
constrained by sensor capabilities and communication loads. In this paper, we
improve V2X perception from road-to-vehicle vision, which utilizes a
stationary Road Side Unit (RSU) (S.Peiyuan 2023; H.TianZhang and et al 2023)
positioned at a fixed location, providing a more stable and expansive sensory
coverage while minimizing redundant communication. As shown in Fig. 1, within
dynamic traffic scenarios, RSU serves as perception nodes situated at the
heart of the traffic scene. This facilitates the collection of comprehensive
perception data over a wider range and ensures more consistent perception as
compared to the inherently mobile vehicles. Furthermore, RSU boasts increased
storage and computing capabilities, enabling rapid adaptation to evolving
environments. _To the best of our knowledge, our work represents a pioneering
effort in harnessing the road-to-vehicle vision paradigm to enhance V2X
perceptions within dynamic scenes._

Motivated by this, this paper proposes Adaptive Road-to-Vehicle Perception
(AR2VP) approach, which constructs a road-to-vehicle cooperative perception
model tailored for dynamic environments. First, to effectively handle intra-
scene changes, we design a dynamic perception representing module and road-to-
vehicle perception compensating module. These modules collaboratively
construct an adaptable graph catering to intra-scene changes, thereby
enhancing vehicles’ overall adaptability within dynamic scenarios. Second, to
effectively adapt to inter-scene changes, we present RSU experience replay,
utilizing RSU storage capacity and integrating experience replay (T.Ren 2023;
M.van and S.Tolias 2020; L.Hao and C.Xiong 2019) techniques in continual
learning (L.Fan and et al 2023; S.Qing and et al 2023; C.Hao 2023; D.Kaile and
et al 2023; L.Fan and et al 2021), enabling vehicles to adapt to large-scale
scene transitions beyond intra-scene changes. Our approach is validated on the
tasks of scene segmentation and 3D object detection. The results on V2X-Sim
dataset (Y. Li and Wang 2022) show good perception performance and the
adaptability of AR2VP to dynamic scenes.

Our contributions are three-fold:

  * •

To the best of our knowledge, we are the first to investigate the adaptability
of V2X to dynamic scenes. Accordingly, we also proposes AR2VP approach for
dynamic V2X perception.

  * •

To address intra-scene changes, we design Dynamic Perception Representing
module and Road-to-Vehicle Perception Compensating module. These modules tap
into the perceptual insights from the road side, thereby bolstering the
overall adaptability of vehicles within dynamic scenes.

  * •

To effectively handle intra-scene changes, we put forward the concept of RSU
Experience Replay. This mechanism empowers vehicles to seamlessly adapt to
substantial scene transitions that extend beyond the scope of mere intra-scene
changes.

## Related Work

Perception in V2X. V2X technology encompasses various forms of cooperative
communication, including Vehicle-to-Vehicle (V2V) (A.Demba and D.P.F.Möller
2018) and Vehicle-to-Infrastructure (V2I) (Ha.Wang and Y.Cai 2022). For V2V
technology, Who2com (Y.-C.Liu and N.Glaser 2020b) exploits a handshake
communication mechanism to determine which two vehicles should communicate for
image segmentation. When2com (Y.-C.Liu and N.Glaser 2020a) introduces an
asymmetric attention mechanism to decide when to communicate and how to create
communication groups for image segmentation. V2VNet (R.Xu and J.Ma 2022)
proposes multiple rounds of message passing on a spatial-aware graph neural
network for joint perception and prediction in autonomous driving. DiscoNet
(M.van and S.Tolias 2020) proposes distilled collaboration graph with matrix-
valued edge weights for adaptive perception, offering superior performance-
bandwidth trade-off. V.Nicholas and et al (2020) proposes a pose error
regression module to learn to correct pose errors when the pose information
from other vehicles is noisy. For V2I technology, the collaboration is between
infrastructure and vehicles, which expands the vehicle’s perception field.
_However, most of both existing V2V and V2I methods build V2X perception model
from vehicle vision, which is insufficient in dynamic traffic environment._ In
this paper, we aim to construct road-to-vehicle vision to address the
challenge of inadequate adaptability of collaborative perception models in
dynamic environments.

Continual Learning. Continual learning is a commonly used approach for
adapting to changing scenarios. It allows the model to continuously update
itself while receiving new data, thereby accommodating various environmental
changes. Some common methods in this context include regularization (H.Kai and
G.Yutao 2023; D.Jiahua and S.Gan 2023), experience replay, and parameter
freezing (W.Chenglong and et al 2023; X.Guangkai and et al 2023). Traffic
scenes are characterized by significant variability, where continual learning
holds promise for application in complex and dynamic traffic environments.
_However, despite its successes in other domains, there is currently no
research considering the utilization of continual learning for modeling V2X
perception systems._ Traditional V2X technologies did not account for scene
changes, resulting in the perception model experiencing forgetting phenomena
(G.Winata and et al 2023; C.Shao and et al 2022) when vehicles transition
between different scenes. This paper investigates the potential of applying
continual learning to model V2X perception systems, with the aim of better
adapting to inter-scene changes, thereby enhancing the safety and reliability
of the automomous.

Figure 2: Overall perception model framework. For the vehicle (green), its BEV
map ${\mathbf{V}_{1}}$ is encoded by the shared Encoder to obtain the feature
map ${\mathbf{M}_{1}}$. Based on the Dynamic Perception Representing module,
the neural information from other vehicles and RSU is aggregated to obtain the
feature map ${\mathbf{M}_{1}^{\rm d}}$. It is compensated with the neural
information ${\mathbf{M}_{0}}$ from the RSU to obtain the feature map
${\mathbf{M}_{1}^{\rm c}}$. The shared header outputs the result after the
shared Decoder.

## Adaptive Road-to-Vehicle Perception

### Overview

We study the V2X perception task with RSU placed on dynamic scenes. In one
scene, the V2X perception consists of an RSU and vehicles. The RSU and
vehicles collect point cloud data, these input single-view point cloud can be
converted to bird’s-eye-view (BEV) (M.van and S.Tolias 2020) maps
${\mathcal{V}}={\\{\mathbf{V}_{0},\mathbf{V}_{1},\mathbf{V}_{2},...,\mathbf{V}_{i}\\}}$,
where ${\mathbf{V}_{0}}$ for RSU and ${\mathbf{V}_{i}}$ for the $i$-th vehicle
($i>0$). Existing V2X perception technique primarily focus on static data,
which falls short of meeting the safety requirements in dynamic traffic
environments: Intra-scene changes, such as pedestrians in motion and moving
vehicles, along with inter-scene changes like transitions between extensive
structures and road layouts across different locations, introduce disruptions
to V2X perception, potentially compromising vehicle safety.

Motivated by this, this work considers to build a collaborative perception
model from road-to-vehicle vision for sensing complex and dynamic traffic
scenarios. We name the method Adaptive Road-to-Vehicle Perception (AR2VP), as
shown in Fig. 2, where vehicles and RSU communicate and cooperate through a
broadcast communication channel. AR2VP considers to address two kinds of scene
changes: Intra-scene changes: we first design _Dynamic Perception
Representing_ (DPR) module, utilizing RSU geographical and perceptual
advantages to effectively integrate the perception from vehicles, enabling
vehicles to capture a more comprehensive range of dynamic factors within the
scene. Then, to further enhance vehicles perception capabilities in dynamic
environments, we draw inspiration from residual (Z.Lei and et al 2023; D.Wenlu
and et al 2023) techniques and propose _Road-to-Vehicle Perception
Compensating_ (R2VPC) module. Leveraging RSU perceptual advantages, this
module compensates the post-collaborative perception of vehicles, filling in
intra-scene dynamic factors that overlooked by the vehicles, thereby further
enhancing the overall adaptability of vehicles to dynamic environments.
Lastly, to enable vehicles to adapt to large-scale scene transitions beyond
intra-scene changes. Inter-scene changes: we introduce _RSU Experience
Replay_. This combines RSU storage capability with experience replay
techniques from continual learning, enabling AR2VP to adapt to inter-scene
changes, ensuring reliable vehicles perception.

### Overcoming intra-scene changes

Dynamic Perception Representing. Vehicle perception varies with the changes of
dynamic entities within the intra-scene. To effectively coordinating these
dynamic factors, this paper propose a Dynamic Perception Representing (DPR)
module, which constructs a directed collaborative graph
$\mathcal{G}=\\{\mathcal{M},\mathcal{\xi}\\}$ that leverages the advantages of
RSU to adapt to intra-scene changes (See Fig. 3), where
$\mathcal{M}={\Phi_{\rm shared}}(\mathcal{V})$ is encoded by the shared
encoder ${\Phi_{\rm shared}(\cdot)}$ to generate the feature maps and
$\mathcal{V}$ represents BEV maps. The collaborative graph has three stages
for dynamic perception representation:

_(1) Stage S1: position information transforming_. In this stage, each vehicle
transfers the position to the RSU for interaction. RSU and each vehicle has
its own independent position
${\mathcal{P}}=\\{(x_{0},y_{0}),(x_{1},y_{1}),(x_{2},y_{2}),...,(x_{n},y_{n})\\}$.
When selecting the position of the ${i}$-th vehicle for collaboration, we need
to transform the position information of RSU $(x_{0},y_{0})$ into $(x_{0\to
i},y_{0\to i})$ corresponding to the ${i}$-th vehicles using the position
matrix:

$x_{0\to i}={\mathbf{R}_{i}}{\mathbf{R}_{0}^{\top}}(x_{0}-x_{i}),\quad y_{0\to
i}={\mathbf{R}_{i}}{\mathbf{R}_{0}^{\top}}(y_{0}-y_{i}),$ (1)

where ${\mathbf{R}_{i}}$ denotes the rotation matrix of the ${i}$-th vehicle,
which represents the orientation of coordinate system relative to the
reference coordinate system. Note that ${\mathbf{R}_{0}}$ is RSU’s rotation
matrix. Then, we pass the converted position information to the S2 stage to
obtain the edge weights.

Figure 3: Dynamic Perception Representing. For the feature map
$\mathbf{M}_{1}$, we calculate the weights $\mathcal{\xi}$ by combining
feature similarity with the RSU-Vehicle distances
$\mathcal{D}=\\{d_{1},d_{2},...,d_{i}\\}$ to construct a directed
collaborative graph $\\{\mathcal{M},\mathcal{\xi}\\}$, followed by perceptual
representation of the feature map $\mathbf{M}_{i}^{\rm d}$.

_(2) Stage S2: Position-guided feature fusing_. In this stage, each vehicle
receives effective perception information from both RSU and other vehicles in
the same scene. RSU, due to its unique geographical position, offers vehicles
dynamic environmental adaptability. Therefore, based on the position
information from Stage S1, we combine the relative distance between RSU and
vehicles with the feature information between vehicles, and carry out
effective collaborative perception. In the directed collaborative graph
$\mathcal{G}$, to determine edge weights $\mathcal{\xi}$, we first obtain the
distances $\mathcal{D}=\\{d_{1},d_{2},...,d_{i}\\}$ between vehicles and the
RSU from the Stage S1:

${d_{i}}=\sqrt{{(x_{0\to i}-x_{i})}^{2}+{(y_{0\to i}-y_{i})}^{2}}.$ (2)

Then, we associate the features of different vehicles. In other words, the
matrix value of the edge weight from the ${2}$-th vehicle to the ${1}$-th
vehicle $\xi_{2\to 1}$:

$\xi_{2\to 1}=\frac{d_{2}\cdot{\rm
cos}({\mathbf{M}_{1}},{\mathbf{M}_{2}})}{\sum\nolimits_{i=2}^{N}{d_{i}\cdot{\rm
cos}({\mathbf{M}_{1}},{\mathbf{M}_{i}})}},$ (3)

where ${\rm norm(\cdot)}$ represents set normalization, and ${\rm cos(\cdot)}$
represents feature similarity. For the edge weights between RSU and vehicles,
we use fixed weight $\lambda=\frac{1}{N}$ to retain the stable perception
information of RSU. Through Stage 2, we complete the construction of graph
$\mathcal{G}$.

_(3) Stage S3: feature information aggregating_ : In this stage, we utilize
the directed collaborative graph constructed in Stage S2 to synergistically
enhance the representation of each vehicle. The perception information of each
vehicle and RSU is integrated to better capture dynamic entities, achieving a
comprehensive perception of the entire environment. Specifically, each vehicle
aggregates the normalized edge-weighted features of all other vehicles. The
updated feature map of the ${i}$-th vehicles is ${\hat{\mathbf{M}}_{i}}$:

${\hat{\mathbf{M}}_{i}}=\sum\nolimits_{j=1}^{N}{\xi_{j\to
i}}{\mathbf{M}_{i}}+\lambda{\mathbf{M}_{0}}.$ (4)

In the DPR module, this study leverages the perceptual and geographical
advantages of RSU to assist vehicles in perception fusion. This approach
enables the perception model to initially adapt to dynamic environments,
achieving a comprehensive perception effect.

Figure 4: Road-to-Vehicle Perception Compensating. For the feature map
$\mathbf{M}_{1}^{\rm d}$, utilizing RSU-vehicles similarity ratio
$\mathcal{R}$ and RSU perception threshold $\lambda$ to determine the extent
of RSU perception compensation within $\mathbf{M}_{1}^{\rm d}$, thus obtaining
the compensated feature map $\mathbf{M}_{1}^{\rm c}$. Table 1: Segmentation
comparison of intra-scene. The best options are in bold and ${*}$ indicates
the pose-aware version.

Method | Unlabeled | Vehicles | Sidewalk | Ground | Road | Buildings | Pedestrian | Vegetation | mIoU(%)
---|---|---|---|---|---|---|---|---|---
Early Fusion | 65.96 | 90.87 | 94.67 | 94.52 | 97.37 | 94.89 | 50.45 | 90.26 | 84.87
Late Fusion | 48.56 | 72.17 | 86.88 | 85.96 | 93.48 | 85.92 | 18.21 | 80.07 | 71.41
When2com | 41.25 | 65.47 | 69.62 | 58.83 | 83.65 | 62.36 | 27.18 | 62.00 | 58.79
When2com* | 41.42 | 63.47 | 72.19 | 58.81 | 81.02 | 68.55 | 28.18 | 74.36 | 59.75
Who2com | 42.25 | 66.47 | 70.62 | 59.83 | 84.65 | 63.36 | 28.18 | 63.00 | 59.80
Who2com* | 40.02 | 63.47 | 72.60 | 62.81 | 81.00 | 60.55 | 28.20 | 66.36 | 60.75
V2V | 60.10 | 84.92 | 93.04 | 91.87 | 95.98 | 93.10 | 33.89 | 86.85 | 79.01
Disco | 61.15 | 84.75 | 92.82 | 92.62 | 96.52 | 92.95 | 35.49 | 87.01 | 80.41
AR2VP | 98.89 | 85.31 | 93.37 | 92.86 | 96.63 | 93.31 | 33.63 | 86.38 | 85.05

Road-to-Vehicle Perception Compensating. Due to the continuous changes of
scenes, using only the collaborative graph for vehicle perception in dynamic
scenarios is insufficient. This paper further leverages the advantages of RSU
perceptual stability and extensive coverage to compensate for the updated
vehicles perception, thereby enhancing vehicles perception in dynamic scenes.
At this stage, our objective is to utilize the perception features of RSU to
compensate for the updated feature maps of vehicles during the decoding
process (See Fig. 4).

First, we flat the feature maps of the RSU and vehicles:

${\mathbf{F}^{i}}={\rm flatten}({\mathbf{M}}_{i}^{\rm d}),$ (5)

where ${\mathbf{M}}_{i}^{\rm d}$ is obtained by decoded from
${\hat{\mathbf{M}}_{i}}$ using a fully-connected layer. Then, we calculate the
feature similarity ratio (Pearson Correlation Coefficient (S.Anuradha and et
al 2023)) ${\mathcal{R}}=\\{r_{1},r_{2},...,r_{i}\\}$ between the RSU and
vehicles, which is to accurately determine which feature map needs to be
compensated using RSU perception, avoiding unnecessary computational burdens.
The formula is as follows:

${r_{i}}=\frac{\sum\nolimits_{i=1}^{n}{a_{i}}}{\sqrt{\sum\nolimits_{i=1}^{n}{a_{0}^{2}}\cdot{a_{i}^{2}}}},$
(6)

where

${a_{i}}=\sum\nolimits_{j=1}^{n}({\mathbf{F}_{j}^{i}}-\sum\nolimits_{i=1}^{k}\frac{\mathbf{M}_{i}^{d}}{k}).$
(7)

Then, a predefined threshold $\lambda$ is used to determine whether
compensation with RSU is required for feature mappings. We supplement them
with RSU perception information, resulting in compensated feature maps
${\mathcal{M}^{\rm c}}$:

${\mathbf{M}}_{i}^{\rm c}=(\lambda-{r_{i}}){\mathbf{M}}_{0}^{\rm
d}+{\mathbf{M}}_{i}^{\rm d}.$ (8)

Finally, the model is followed by different output heads based on the tasks
(such as segmentation (seg.) and detection (det.)) to generate perception
results. Though AR2VP adapts to the changes of intra-scene, but adapting to
the changes of inter-scene is still unsolved.

Figure 5: RSU Experience Replay.

### Overcoming inter-scene changes

The change of inter-scene is relatively steep, and when the model learns a new
scene, it will lead to the forgetting of old scenes knowledge. Inspired by
this, this paper introduces RSU Experience Replay (RSU-ER), leveraging RSU
storage advantage to retain old scene data and applying continual learning
techniques to mitigate forgetting (See Fig. 5).

First, at the learning on new scene data ${\mathcal{S}_{\rm c}}$, we utilize
the storage capability of RSU to store a small set of samples
${\mathcal{S}_{\rm p}}$:

${\mathcal{S}_{\rm p}}={\rm Select}(\mu,{\mathcal{S}_{\rm c}}),$ (9)

where, ${\rm Select}(\mu,\cdot)$ represents random selection operation, and
$\mu$ is the selected number. Then, as the model learns new scenes again, we
randomly extract a small portion of samples from the ${\mathcal{S}_{\rm p}}$,
concatenate them with the new scene samples for model updating:

$\theta={\rm SGD}({\mathcal{S}_{\rm c}}\cup{\mathcal{S}_{\rm p}},\theta),$
(10)

where, ${\rm SGD}(\cdot)$ is the stochastic gradient descent algorithm, and
$\theta$ represents the model parameters. Lastly, once the model updating is
completed, we refresh the ${\mathcal{S}_{\rm p}}$ once more:

${\mathcal{S}_{\rm p}}={\rm Select}(\mu,{\mathcal{S}_{\rm
c}})\cup{\mathcal{S}_{\rm p}}.$ (11)

In this stage, the model engages in comprehensive learning of the previous
data stored in RSU and the current scene data. This not only acquires
knowledge from new scenes but also revisits knowledge from previous scenes,
achieving a synergistic combination of learning and review. Our RSU-ER
effectively mitigates the catastrophic forgetting caused by inter-scene
changes.

### The whole algorithm

In the process of model learning update to dataset ${\mathcal{S}}$, we use
${L_{\rm det}}$ loss for the detection (E.Verwimp and et al 2022; L.Sebastian
and P.A.Hooper 2023) task to update learning:

$L_{\rm
det}=\sum\nolimits_{i=1}^{n}{\frac{\eta({Y_{i}-Y_{i}^{\prime}})^{2}}{\sigma^{2}}},$
(12)

where, $\eta$ typically takes a value of 0.5, in the segmentation (C.Yiming
2023; H.Xie and et al 2023) task, we use ${L_{\rm seg}}$ loss for update
learning:

$L_{\rm seg}=-\sum\nolimits_{i=1}^{n}(Y_{i}\cdot\log(Y_{i}^{\prime})),$ (13)

where ${Y}$ and ${Y^{\prime}}$ represent the label and prediction in scene
${\mathcal{S}}$, ${\sigma}$ is a hyperparameter.

The overall update loss ${L}$ of the model is as follows:

$L=L_{\rm det/seg}^{\rm previous}+L_{\rm det/seg}^{\rm current},$ (14)

where $L_{\rm det/seg}^{\rm previous}$ represents the loss from replayed
previous scene data used for the current task $(\rm det./seg.)$, and $L_{\rm
det/seg}^{\rm previous}$ represents the loss from current scene data used for
the current task.

In the whole AR2VP research (See Fig. 2), we design DPR module (See Fig. 3),
merging geographical and feature data from RSU and vehicles to create an
adaptable collaborative graph for dynamic scenarios. This effectively
integrates perception information from different vehicles, enabling a more
comprehensive grasp of dynamic elements within the scene. Subsequently,
inspired by residual techniques, we propose the R2VPC module (See Fig. 4). By
leveraging RSU perceptual advantages, this module compensates post-
collaborative vehicle perception, filling in intra-scene dynamic elements
overlooked by the vehicles, further enhancing overall adaptability to dynamic
settings. Lastly, to extend adaptability beyond intra-scene changes, we
introduce RSU-ER (See Fig. 5), combining RSU storage capacity and experience
replay techniques. This empowers AR2VP to cope with inter-scene changes,
ensuring robust and reliable vehicle perception. Note that AR2VP also
considers _to save the communication bandwidth_ , where RSU and vehicles could
compress their feature map prior to transmission. Optionally, in our study, we
make use a 1 × 1 convolutional autoencoder (M.Jonathan and S.Jürgen 2011) to
compress and decompress the feature maps along the channel dimension. The
autoencoder is trained together with the whole system.

Figure 6: Visualizations of collaborative BEV semantic segmentation. Figure 7:
Visualizations of BEV detection on V2X-Sim. Red and green boxes are the
predictions and ground-truths respectively.

## Experiment

### Data preparation and evaluation metric

In this study, we employ the V2X-sim dataset to evaluate the V2X perception
task. The V2X-sim dataset emulates multi-agent scenarios, wherein each
scenario encompasses a 20-second traffic flow across multiple intersections.
Laser radar recordings are captured at intervals of 0.2 seconds, yielding a
total of 100 frames per scenario. This dataset comprises 100 distinct scenes,
with each frame housing multiple samples. The training set comprises 23,500
samples, while the test set contains 3,100 samples. To establish a fixed large
scenario, we selected 30 scenes, which collectively contribute 3,000 frames.
Among these, the training set comprises 2,700 frames, while the test set
consists of 300 frames. Moreover, to implement cross-scene experiments, we
also train V2X model sequentially on three major scenes in chronological
order.

Table 2: Detection comparison of intra-scene.

Method | mAP(%)
---|---
<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
Early Fusion | 96.63 | 96.05
Late Fusion | 85.62 | 83.84
When2com | 81.35 | 80.02
When2com* | 81.86 | 80.69
Who2com | 81.32 | 79.98
Who2com* | 81.69 | 80.66
V2V | 91.89 | 89.90
Disco | 92.01 | 90.41
AR2VP | 94.50 | 92.77

In this paper, we evaluate our method on two V2X perception tasks, including
scene segmentation and vehicle objection. We employ the generic BEV detection
evaluation metric: Average Precision (AP) at Intersection-over-Union (IoU)
threshold of 0.5 and 0.7. We evaluate the segmentation performance using mean
IoU (mIoU). We evaluate the extent of forgetting across inter-scene changes
using Forget.

### Quantitative evaluation

Compared methods. We first compare with the early collaboration method (C.Qi
and F.Song 2019) and the late collaboration method, which are always seems as
the upper bound and lower in traditional V2X perception tasks. Furthermore,
four intermediate collaboration methods are used, inculuding When2com
(Y.-C.Liu and N.Glaser 2020a), Who2com (Y.-C.Liu and N.Glaser 2020b), V2V
(T.Wang and et al 2020) and Disco (L.Yiming and et al 2021). Since the
original Who2com and When2com do not consider pose information, we consider
both pose-aware and pose-agnostic versions (with $*$) to achieve fair
comparisons. All the methods use the same segmentation and detection backbones
and conduct collaboration at the same intermediate feature layer.

Comparisons under intra-sence changes. Tables 1 and 2 show the comparisons in
terms of mIoU (seg.) and<EMAIL_ADDRESS>(det.). Comparing to the pose-aware
When2com, AR2VP improves by 57.57% in segmentation of unlabeled data and
26.26% in mIoU. Comparing to Disco, AR2VP improves by 4.64%. Comparing to the
pose-aware When2com, AR2VP improves by 13.15% in<EMAIL_ADDRESS>and 12.75% in<EMAIL_ADDRESS>Comparing to Disco, AR2VP improves by 3.4% in<EMAIL_ADDRESS>and 3.2% in<EMAIL_ADDRESS>The
qualitative results are shown in Fig. 6 (seg.) and Fig. 7 (det.). We observed
that AR2VP demonstrates superior entity perception outcomes, achieving the
highest overall perception performance. This analysis underscores that current
V2X technologies rarely rely on RSUs to expand perception horizons. In
contrast, AR2VP harnesses the latent strengths of RSUs to address intra-scene
changes, which enhances the vehicle’s ability to adapt to dynamic scenes,
consequently elevating the overall perception capabilities. However, AR2VP
does exhibit a performance drawback in pedestrian detection, implying a
particular challenge in detecting small targets.

Comparisons under inter-scene change. Table 3 shows the comparison on
inhibition of forgetting cross different scenes. Comparing to the pose-aware
When2com, AR2VP improves by 30.78% in mIoU and reduce forgetting rate by
19.28%. Comparing to Disco, AR2VP improves by 20.42% in mIoU and reduce
forgetting rate by 23.42%; Comparing to the pose-aware When2com, AR2VP
improves by 28.12% in<EMAIL_ADDRESS>and 27.00% in<EMAIL_ADDRESS>and reduce forgetting rate
by 26.07% in<EMAIL_ADDRESS>and 25.60% in<EMAIL_ADDRESS>Comparing to Disco, AR2VP improves
by 6.05% in<EMAIL_ADDRESS>and 6.44% in<EMAIL_ADDRESS>and reduce forgetting rate by 8.23% in
<EMAIL_ADDRESS>and 9.94% in<EMAIL_ADDRESS>AR2VP presents itself as a frontrunner in terms of
overall perception performance. Upon analysis, it’s evident that traditional
V2X technologies disregard the influence of inter-scene changes on perception.
In contrast, AR2VP optimally exploits the storage capacity of RSU and
integrates continuous learning principles to effectively address inter-scene
changes. This strategic approach empowers vehicles to assimilate new scenes
while minimizing the extent of memory loss from prior scenes. This capability
shows a strong adaptability to inter-scene changes in perception, thereby
enhancing the global robustness of perception. Although RSU-ER can be applied
to other models to mitigate forgetting, AR2VP notably demonstrates the most
favorable suitability. Fig. 8 portrays the learning of new scene data based on
the old model, revealing the degree of memory forgetting from previous scenes.
The observation is clear: V2V and Disco struggle to accommodate inter-scene
changes, leading to significant memory loss from previous scenes. In contrast,
AR2VP adeptly navigates inter-scene changes, exhibiting a higher retention of
memories from prior scenes. This analysis underscores AR2VP’s capacity for the
lowest forgetting rate and the most proficient performance in addressing
inter-scene changes.

Figure 8: Visualizations of collaborative BEV semantic segmentation on
inhibition of forgetting.  Figure 9: Performance-bandwidth trade-off.

Performance-bandwidth trade-off analysis. In Fig. 9, we compare the proposed
AR2VP with the baseline methods in terms of the trade-off between segmentation
performance and communication bandwidth. The dashed line represents the
baseline based on when2com. To show the better trade-off of the proposed
AR2VP, we employ an autoencoder to compress features and reduce the
communication bandwidth used for feature transmission ($/n$ means compress $n$
times). We have the following observations: 1) Comparing to AR2VP, AR2VP/32
degrades by 1.31% in segmentation, still outperforming Disco in terms of mIoU.
This means the compressing features in AR2VP does not significantly compromise
perception performance. 2) Storing the model in the RSU further reduces
${n^{2}}$ communication bandwidth, where ${n}$ is the number of vehicles
participating in the collaboration.

Table 3: Perception comparisons on inter-scene changes.

Method | Det.(AP@0.5(%)) | Det.(AP@0.7(%)) | Seg.(%)
---|---|---|---
mAP↑ | Forget↓ | mAP↑ | Forget↓ | mIoU↑ | Forget↓
Early Fusion | 85.15 | 14.96 | 81.03 | 19.19 | 52.71 | 42.62
Late Fusion | 62.77 | 27.61 | 57.86 | 31.96 | 4014 | 31.91
When2com | 61.40 | 32.23 | 57.90 | 34.44 | 40.73 | 30.76
When2com* | 61.64 | 31.38 | 57.98 | 33.01 | 45.60 | 28.65
Who2com | 61.04 | 32.89 | 57.66 | 34.63 | 41.63 | 31.56
Who2com* | 61.47 | 32.01 | 58.11 | 33.79 | 44.71 | 29.56
V2V | 82.54 | 15.65 | 76.98 | 20.63 | 48.50 | 46.08
Disco | 83.47 | 14.39 | 78.46 | 18.78 | 48.09 | 43.31
V2V (RSU-ER) | 87.63 | 7.38 | 83.48 | 9.46 | 63.06 | 18.09
Disco (RSU-ER) | 87.65 | 7.61 | 83.65 | 9.53 | 64.74 | 19.12
AR2VP (RSU-ER) | 89.52 | 6.16 | 84.90 | 8.84 | 71.51 | 11.48

Table 4: Ablation studies.

RSU | graph | compensator | Det.(%) | Seg.(%)
---|---|---|---|---
<EMAIL_ADDRESS>|<EMAIL_ADDRESS>| mIoU
✗ | ✗ | ✓ | 66.99 | 65.47 | 55.01
✗ | ✓ | ✗ | 89.88 | 87.95 | 75.65
✗ | ✓ | ✓ | 90.65 | 88.46 | 73.06
✓ | ✗ | ✓ | 68.56 | 66.32 | 56.47
✓ | ✓ | ✗ | 93.80 | 91.71 | 84.04
✓ | ✓ | ✓ | 94.50 | 92.77 | 85.05

Ablation study. We conduct ablation studies to analyze the perceptual
performance of graph, and the communication in the presence and absence of
RSU. The results are shown in Table 4. First, we find that the participation
of RSU in the collaborative process provide additional perception coverage to
enhance vehicle perception performance. Second, the collaborative graph
effectively integrates all perception information, enabling vehicles to
comprehensively perceive entities within the scene. Third, in scenarios where
RSU is present, the compensator utilize the stable perception information from
RSU to efficiently compensate for vehicle perception. Moreover, the
compensator benefits from the presence of RSU, showing a positive effect. In
the absence of RSU, using vehicles to compensate for other vehicles’
perception would lead to negative consequences.

## Conclusion

In this paper, we proposed a vehicle-road cooperative perception model, named
AR2VP, which is capable of adapting to dynamic environments. It mainly
consists of a DPR module, a R2VPC module and RSU-ER method. The DPR module
efficiently integrates vehicle perceptions to comprehensively capture dynamic
factors within the scene, enhancing the perception capabilities of the
collaborative perception model. The R2VPC module is geared towards effectively
retaining the optimal RSU perception information, especially in the face of
intra-scene changes. The RSU-ER method integrates within the RSU’s storage
capacity, facilitates the retention of a small volume of historical scene
data. This approach ensures that the cooperative model maintains a certain
level of robustness when confronted with inter-scene changes. Comprehensive
experiments demonstrate that AR2VP achieves adaptability to dynamic
environments and an appealing performance-bandwidth trade-off through a more
direct design principle. In the future, based on our experimental findings, we
intend to enhance the AR2VP’s capability in recognizing small objects. This
will involve further refinement of the model to ensure more accurate and
effective identification of small entities.

## References

  * A.Demba and D.P.F.Möller (2018) A.Demba; and D.P.F.Möller. 2018.  Vehicle-to-Vehicle Communication Technology.  In _EIT_.
  * A.Geiger (2012) A.Geiger, R., P.Lenz. 2012.  Are we ready for autonomous driving? the KITTI vision benchmark suite.  In _CVPR_.
  * C.Hao (2023) C.Hao, L., L.LinYan. 2023.  Multi-semantic hypergraph neural network for effective few-shot learning.  _PR_.
  * C.Qi and F.Song (2019) C.Qi, Y., T.Sihai; and F.Song. 2019.  Cooper: Cooperative perception for connected autonomous vehicles based on 3d point clouds.  In _ICDCS_.
  * C.Shao and et al (2022) C.Shao, Y.; and et al. 2022.  Overcoming Catastrophic Forgetting beyond Continual Learning: Balanced Training for Neural Machine Translation.  In _ACL_.
  * C.Yiming (2023) C.Yiming, Y., Y.Linjie. 2023.  Learning Dynamic Query Combinations for Transformer-based Object Detection and Segmentation.  In _ICML_.
  * D.Jiahua and S.Gan (2023) D.Jiahua, C., L.Wenqi; and S.Gan. 2023.  Heterogeneous Forgetting Compensation for Class-Incremental Learning.  In _ICCV_.
  * D.Kaile and et al (2023) D.Kaile, L.; and et al. 2023.  Multi-Label Continual Learning using Augmented Graph Convolutional Network.  _TMM_.
  * D.Wenlu and et al (2023) D.Wenlu, G., Y.Junyi; and et al. 2023.  SafeLight: A Reinforcement Learning Method toward Collision-free Traffic Signal Control.  In _AAAI_.
  * E.Verwimp and et al (2022) E.Verwimp, S., K.Yang; and et al. 2022.  CLAD: A realistic Continual Learning benchmark for Autonomous Driving.  In _ICLR_.
  * G.Winata and et al (2023) G.Winata, K., L.Xie; and et al. 2023.  Overcoming Catastrophic Forgetting in Massively Multilingual Continual Learning.  In _ACL_.
  * Ha.Wang and Y.Cai (2022) Ha.Wang, X.; and Y.Cai. 2022.  V2I-CARLA: A Novel Dataset and a Method for Vehicle Reidentification-Based V2I Environment.  _IEEE TIM_.
  * H.Kai and G.Yutao (2023) H.Kai, X., W.Feigege; and G.Yutao. 2023.  Prototypical Kernel Learning and Open-set Foreground Perception for Generalized Few-shot Semantic Segmentation.  In _ICCV_.
  * H.TianZhang and et al (2023) H.TianZhang, N., Adel; and et al. 2023.  An Intent-based Framework for Vehicular Edge Computing.  In _PerCom_.
  * H.Xie and et al (2023) H.Xie, K., J.Zhu; and et al. 2023.  A Critical View of Vision-Based Long-Term Dynamics Prediction Under Environment Misalignment.  In _ICML_.
  * L.Fan and et al (2023) L.Fan, S.; and et al. 2023.  Measuring Asymmetric Gradient Discrepancy in Parallel Continual Learning .  In _ICCV_.
  * L.Fan and et al (2021) L.Fan, W.; and et al. 2021.  Multi-Domain Multi-Task Rehearsal for Lifelong Learning.  In _AAAI_.
  * L.Hao and C.Xiong (2019) L.Hao, A.; and C.Xiong. 2019.  Competitive Experience Replay.  In _ICLR_.
  * L.Sebastian and P.A.Hooper (2023) L.Sebastian; and P.A.Hooper. 2023.  In-situ Anomaly Detection in Additive Manufacturing with Graph Neural Networks.  In _ICLR_.
  * L.Yiming and et al (2021) L.Yiming, W., R.Shunli; and et al. 2021.  Learning Distilled Collaboration Graph for Multi-Agent Perception.  In _NIPS_.
  * M.Hasan and H.Lu (2018) M.Hasan, T., S.Mohan; and H.Lu. 2018.  Securing vehicle-to-everything (V2X) communication platforms.  _IEEE Trans. Intell. Veh._
  * M.Jonathan and S.Jürgen (2011) M.Jonathan, C., M.Ueli; and S.Jürgen. 2011.  FrozenRecon: Pose-free 3D Scene Reconstruction with Frozen Depth Models.  In _ICANN_.
  * M.Muhammad and G.A.Safdar (2018) M.Muhammad; and G.A.Safdar. 2018.  Survey on existing authentication issues for cellular-assisted V2X communication.  _Vehicle Communication_.
  * M.van and S.Tolias (2020) M.van, T.; and S.Tolias. 2020.  Brain-inspired replay for continual learning with artificial neural networks.  _Nature Communications_.
  * R.Xu and J.Ma (2022) R.Xu, X., H.Xiang; and J.Ma. 2022.  OPV2V: An open benchmark dataset and fusion pipeline for perception with vehicle-to-vehicle communication.  In _ICRA_.
  * S.Anuradha and et al (2023) S.Anuradha, Y.; and et al. 2023.  Linking Alternative Fuel Vehicles Adoption with Socioeconomic Status and Air Quality Index.  In _AAAI_.
  * S.Peiyuan (2023) S.Peiyuan, Z., Q.Liangxin. 2023.  A Hybrid Framework of Reinforcement Learning and Convex Optimization for UAV-Based Autonomous Metaverse Data Collection.  _IEEE Network magazine_.
  * S.Qing and et al (2023) S.Qing, L.; and et al. 2023.  Exploring Example Influence in Continual Learning.  In _NeurIPS_.
  * T.Ren (2023) T.Ren, Z. 2023.  Integrating Curricula with Replays: Its Effects on Continual Learning.  In _AAAI_.
  * T.Wang and et al (2020) T.Wang, L., S.Manivasagam; and et al. 2020.  V2vnet: Vehicle-to-vehicle communication for joint perception and prediction.  In _ECCV_.
  * V.Nicholas and et al (2020) V.Nicholas, R.; and et al. 2020.  Learning to communicate and correct pose errors.  In _CoRL_.
  * W.Chenglong and et al (2023) W.Chenglong, Z., Y.Jiangyan; and et al. 2023.  Low-rank Adaptation Method for Wav2vec2-based Fake Audio Detection.  In _IJCAI_.
  * X.Guangkai and et al (2023) X.Guangkai, G., Y.Wei; and et al. 2023.  FrozenRecon: Pose-free 3D Scene Reconstruction with Frozen Depth Models.  In _ICCV_.
  * Y.-C.Liu and N.Glaser (2020a) Y.-C.Liu, J.; and N.Glaser. 2020a.  When2com: Multi-agent perception via communication graph grouping.  In _CVPR_.
  * Y.-C.Liu and N.Glaser (2020b) Y.-C.Liu, J.; and N.Glaser. 2020b.  Who2com: Collaborative perception via learnable handshake communication.  In _ICRA_.
  * Y. Li and Wang (2022) Y. Li, Z. A., Dekun Ma; and Wang, Z. 2022.  V2X-Sim: Multi-Agent Collaborative Perception Dataset and Benchmark for Autonomous Driving.  _RAL_.
  * Y.Li and et al (2021) Y.Li, P., S.Ren; and et al. 2021.  Learning distilled collaboration graph for multi-agent perception.  In _NIPS_.
  * Y.Yuan and M.Sester (2021) Y.Yuan; and M.Sester. 2021.  Comap: A synthetic dataset for collective multiagent perception of autonomous driving.  In _NIPS_.
  * Z.Jiarui and et al (2023) Z.Jiarui, K., I.Filip; and et al. 2023.  Utilizing Background Knowledge for Robust Reasoning over Traffic Situations.  In _AAAI_.
  * Z.Lei and et al (2023) Z.Lei, H., Y.Xiaodong; and et al. 2023.  DRGCN: Dynamic Evolving Initial Residual for Deep Graph Convolutional Networks.  In _AAAI_.


#  $S$-packing colorings of distance graphs with distance sets of cardinality
$2$

Boštjan Brešara,b Jasmina Fermec,a Přemysl Holubd Marko Jakovaca,b Petra
Melicharovád

###### Abstract

For a non-decreasing sequence $S=(s_{1},s_{2},\ldots)$ of positive integers, a
partition of the vertex set of a graph $G$ into subsets
$X_{1},\ldots,X_{\ell}$, such that vertices in $X_{i}$ are pairwise at
distance greater than $s_{i}$ for every $i\in\\{1,\ldots,\ell\\}$, is called
an $S$-packing $\ell$-coloring of $G$. The minimum $\ell$ for which $G$ admits
an $S$-packing $\ell$-coloring is called the $S$-packing chromatic number of
$G$, denoted by $\chi_{S}(G)$. In this paper, we consider $S$-packing
colorings of distance graphs $G(\mathbb{Z},\\{k,t\\})$, where $k$ and $t$ are
positive integers, which are the graphs whose vertex set is $\mathbb{Z}$, and
two vertices $x,y\in\mathbb{Z}$ are adjacent whenever $|x-y|\in\\{k,t\\}$. We
complement partial results from two earlier papers, thus determining all
values of $\chi_{S}(G(\mathbb{Z},\\{k,t\\}))$ when $S$ is any sequence with
$s_{i}\leq 2$ for all $i$. In particular, if $S=(1,1,2,2,\ldots)$, then the
$S$-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while
if $S=(1,2,2,\ldots)$, then the $S$-packing chromatic number is $5$, unless
$\\{k,t\\}=\\{2,3\\}$ when it is $6$; when $S=(2,2,2,\ldots)$, the
corresponding formula is more complex.

a Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia

b Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia

c Faculty of Education, University of Maribor, Slovenia

d University of West Bohemia, Plzeň, Czech Republic

Key words: $S$-packing coloring, $S$-packing chromatic number, distance graph,
distance coloring.

AMS Subj. Class: 05C15, 05C12

##  1 Introduction

Given a graph $G$ and a non-decreasing sequence $S=(s_{1},s_{2},\ldots)$ of
positive integers, the mapping $f:V(G)\rightarrow[\ell]=\\{1,\ldots,\ell\\}$
is an $S$-packing $\ell$-coloring of $G$ if for any distinct vertices $u,v\in
V(G)$ with $f(u)=f(v)=i$, $i\in\\{1,\ldots,\ell\\}$, the distance between $u$
and $v$ in $G$ is greater than $s_{i}$. The smallest $\ell$ such that $G$ has
an $S$-packing $\ell$-coloring is the $S$-packing chromatic number of $G$,
denoted by $\chi_{S}(G)$. This concept was introduced by Goddard, Hedetniemi,
Hedetniemi, Harris, and Rall [13], and was studied in a number of papers; see
the recent survey [3] and the references therein. The main focus of the
seminal paper and a number of subsequent papers was on the specific sequence
$S=(n)_{n\geq 1}$ in which positive integers appear in the natural order,
where the resulting graph invariant is simply called the packing chromatic
number [4]. Goddard and Xu [14] started consideration of various non-
decreasing sequences $S$, and a number of authors followed them. Arguably the
most interesting sequences are those that involve only integers $1$ and $2$,
since they are in a sense between standard coloring, where $S$ is the constant
sequence of $1$s, and $2$-distance coloring, where $S$ is the constant
sequence of $2$s (note that a $2$-distance coloring is equivalent to a
coloring of the square of a graph and it has been intensively studied in the
last decades [17]). In particular, $S$-packing colorings of subcubic graphs
were investigated with respect to such sequences $S$ [5, 12, 18]. Roughly a
decade ago, Ekstein et al. [10] and Togni [19] initiated the study of
$S$-packing colorings in integer distance graphs, which we present next. Their
study was motivated by a series of papers on proper vertex coloring of
distance graphs, see e.g. [6, 7, 8, 9] and references therein.

Given a set $D=\\{d_{1},\dots,d_{h}\\}$, $h\geq 1$, of positive integers, the
(integer) distance graph, $G(\mathbb{Z},D)$, is the infinite graph with
$\mathbb{Z}$ as the vertex set, while vertices $x$ and $y$ are adjacent if
$|x-y|\in D$. That is, two vertices/integers are adjacent in the graph if
their distance in $\mathbb{Z}$ is one of the integers in
$\\{d_{1},\ldots,d_{h}\\}$. We will simplify the notation, and instead of
$G(\mathbb{Z},D)$ write $G(D)$, and for distance sets with two integers we
will write $D=\\{k,t\\}$, and always assume that $k<t$; thus the corresponding
distance graph will be written as $G(k,t)$. The packing chromatic numbers of
distance graphs $G(k,t)$ were investigated in [11]. In addition, the
$S$-packing colorings of distance graphs $G(k,t$) where $k\in\\{1,2\\}$ and
$t$ is arbitrary were studied in [16, 2]. Concerning the sequences $S$ which
involve only integers that are not greater than $2$, exact values for
$\chi_{S}(G(1,t))$, where $t\geq 2$, were determined in [16, 1], while the
values $\chi_{S}(G(2,t))$, where $t\geq 3$, were established in [2]. Hence,
for this type of sequences $S$, the $S$-packing chromatic numbers of $G(k,t)$
were left open when $k\geq 3$, and the main goal of this paper is to establish
these remaining values.

In the next section, we establish the notation and give some preliminary
observations. In particular, we present two main tools that are used in the
proofs. First, we present a representation of the graph $G(k,t)$ in the so-
called shifted grid. Second, we introduce color patterns and shift sequences,
which enable us a relatively brief presentation of colorings. In Section 3, we
follow with proving the main results, which are the values of
$\chi_{S}(G(k,t))$, for all $3\leq k<t$, and for all possible sequences $S$
involving only integers $1$ and $2$. In Section 4, we give an overview of
results on $S$-packing colorings of the graph $G(k,t)$ by combining the
results from [16, 2] with new results from this paper.

##  2 Notation and preliminaries

When presenting sequences $S$, we will often use $i^{p}$, where $i$ and $p$
are positive integers, as a shortened notation of the (sub)sequence
$(i,\ldots,i)$, where $i$ appears $p$ times. For instance, $(1^{2},2^{5})$
stands for the (sub)sequence $(1,1,2,2,2,2,2)$. We may also write $i^{\infty}$
which coincides with the infinite (sub)sequence $(i,i,i,...)$. In the case of
distinct integers in the sequence $S$, the integer with the power to infinity
is the largest among the integers in $S$. For instance, $(1^{2},2^{\infty})$
presents the sequence with two integers $1$ and all other integers $2$.

Note that $G(k,t)$ is connected if and only if $\gcd(k,t)=1$. Thus, when
determining the $S$-packing chromatic numbers, we restrict to graphs $G(k,t)$
such that $k$ and $t$ are coprime integers. Note that if $\gcd(k,t)=g$, then
$G(k,t)$ consists of connected components all of which are graphs
$G(\frac{k}{g},\frac{t}{g})$, implying that
$\chi_{S}(G(k,t))=\chi_{S}(G(\frac{k}{g},\frac{t}{g}))$.

During our study we make use of the following presentation. Notably, a
connected distance graph $G(k,t)$ can be represented by the square lattice
$\\{0,1,\dots,t\\}\times\mathbb{Z}$ with vertices given by points, which are
ordered pairs $(i,j),i\in\\{0,1,\dots,t\\},j\in\mathbb{Z}$, such that
vertex/point $(i,j)$ of the grid is a representative of the integer $j\cdot
t+i\cdot k$ from $G(k,t)$. These representatives are unique with the exception
of the points whose first coordinate equals to $0$ or $t$, since a point
$(0,j)$ on the grid represents the same integer of $G(k,t)$ as the point
$(t,j^{\prime})$, where $j^{\prime}=j-k$. See Fig. 1, where ordered pairs in
red present points on the grid, while integers in black present the
corresponding integers from $\mathbb{Z}$.

Furthermore, let column $i$ denote the set of vertices
$B_{i}=\\{(i,j):\,j\in\mathbb{Z}\\}$, where $0\leq i\leq t$. As mentioned
earlier, integers from $V(G(k,t))$ of the form $jt,j\in\mathbb{Z}$, are
represented twice on the grid, notably by a vertex in the column $0$ and a
vertex in column $t$. That is, $jt$ is represented by vertex $(0,j)$ as well
as vertex $(t,j-k)$. For instance, points $(0,0)$ and $(t,-k)$ represent $0\in
V(G(k,t))$.

Figure 1: Representation of the distance graph $G(k,t)$ in the square grid.

###  2.1 Color patterns

In this paper, we will often present an $S$-packing coloring $c$ by using
periodic patterns applied on columns $B_{i}$, where $i\in\\{0,\dots,t\\}$. A
periodic pattern of length $d\geq 2$ is a sequence of colors
$[c_{1},\dots,c_{d}]$ (denoted by square brackets), where the colors
$c_{n},n\in[d]$, are not necessarily pairwise distinct. These colors are given
to consecutive vertices within one column and the pattern is applied
downwards. That is, if the coloring $c$ is using a pattern
$P=[c_{1},\dots,c_{d}]$ in the column $B_{x}$ such that $c(x,y)=c_{1}$ for
some $y$, then $c(x,y-n)=c_{n+1}$ for each $n\in[d-1]$. This pattern is then
periodically copied upwards and downwards to cover all the vertices of
$B_{x}$. Thus, $c(x,y+1)=c_{d}$, $c(x,y+2)=c_{d-1}$ and so on.

In order for $c$ to be an $S$-packing coloring of $G(k,t)$, patterns must be
often shifted in consecutive columns. We define the notion of shift sequence
$(p_{i})_{i=0}^{t-1}$ where $p_{i}\in\mathbb{N}_{0}$. To describe it, we also
need the concept of reference point $(i,j)$ in $B_{i}$, which is a unique
point in each column. Without loss of generality we may declare the reference
point in column $B_{0}$ to be $(0,0)$. The integer $p_{i}\geq 0$ represents
the value by which the reference point in $B_{i+1}$ shifts downwards with
respect to the reference point in $B_{i}$. That is, if the reference point in
$B_{i}$ is $(i,j)$, for some $j\in\mathbb{Z}$, then $(i+1,j-p_{i})$ is the
reference point in column $B_{i+1}$. (Note that $p_{i}=0$ means that there is
no shift.) To make the elements of the shift sequence correspond to the
application of periodic patterns to the columns $B_{i}$, we will always assume
that the reference point of each column $B_{i}$ always receives the first
color of the corresponding pattern applied to this column. That is, if a
coloring $c$ is using the pattern $P=[c_{1},\ldots,c_{d}]$ in a column $B_{x}$
with the reference point $(x,j)$ then $c(x,j)=c_{1}$. Thus, the pattern $P$
used together with the reference point $(x,j)$ completely determines the
coloring of points in $B_{x}$.

When shifting the patterns, there are two cases to consider: either columns
$B_{i}$ and $B_{i+1}$ are using the same pattern or they are using different
patterns. First, suppose that a coloring $c$ uses the same pattern
$[c_{1},\ldots,c_{d}]$ in columns $B_{i}$ and $B_{i+1}$. In this case the
integer $p_{i}$ defines the value by which the color $c_{1}$ in $B_{i+1}$ is
shifted with respect to the color $c_{1}$ in $B_{i}$. In other words, if
$c(i,j)=c_{1}$ for some $j\in\mathbb{Z}$, then $c(i+1,j-p_{i})=c_{1}$. This
implies that $p_{i}\neq 0$ (the pattern must be shifted) since the adjacent
vertices must not receive the same color. Next, let $c$ assign a pattern
$[a_{1},\ldots,a_{d_{1}}]$ to the column $B_{i}$ and a pattern
$[b_{1},\ldots,b_{d_{2}}]$ to the column $B_{i+1}$. In this case, the integer
$p_{i}$ defines the value by which the color $b_{1}$ given to the reference
point of $B_{i+1}$ is shifted with respect to the color $a_{1}$ given to the
reference point of $B_{i}$. This means that if $c(i,j)=a_{1}$, where $(i,j)$
is the reference point in $B_{i}$, then $c(i+1,j-p_{i})=b_{1}$ (note that
$(i+1,j-p_{i})$ is the reference point in $B_{i+1}$).

Since the points $(0,j)$ represent the same vertices as points $(t,j-k)$, the
relation $c(0,j)=c(t,j-k)$ must hold for each $j\in\mathbb{Z}$. Thus, if $c$
is an $S$-packing coloring of $G(k,t)$, then columns $B_{0}$ and $B_{t}$ must
use the same periodic pattern and $\sum_{i=0}^{t-1}p_{i}\equiv k\pmod{d}$,
where $d$ is the length of the pattern used on $B_{0}$. Note that it is
sufficient to consider only shifts $p_{i}<d_{\max}$ where $d_{\max}$ denotes
the length of the longest pattern.

In this paper, all colorings given by periodic patterns and shift sequences
will either use one or two different patterns. If the second option occurs, it
is necessary to specify which column receives which pattern. Let $c$ be an
$S$-packing coloring of $G(k,t)$ given by the shift sequence
$(p_{i})_{i=0}^{t-1}$ and two periodic patterns
$P_{1}=[a_{1},\dots,a_{d_{1}}]$ and $P_{2}=[b_{1},\dots,b_{d_{2}}]$ such that
pattern $P_{2}$ is used for example on columns $B_{1}$ and $B_{3}$, and the
rest of the columns obtain pattern $P_{1}$. We will abbreviate such
description as:

$[a_{1},\dots,a_{d_{1}}]_{p_{0}}[b_{1},\dots,b_{d_{2}}]_{p_{1}}[a_{1},\dots,a_{d_{1}}]_{p_{2}}[b_{1},\dots,b_{d_{2}}]_{p_{3}}[a_{1},\dots,a_{d_{1}}]_{p_{4\rightarrow
t-1}}[a_{1},\dots,a_{d_{1}}],$

provided that the shift sequence has $p_{4}=\ldots=p_{t-1}$.

For a better understanding of the concepts, we next provide a more specific
example. Let $c$ be an $(1^{6})$-coloring of $G(k,t)$ given by
$P_{1}=[1,2,3]$, $P_{2}=[4,5,6]$ and the shift sequence $(0,1,1,0,1^{t-4})$
where the notation $1^{t-4}$ stands for the (sub)sequence $(1,\dots,1)$, where
$1$ appears $(t-4)$ times. Again, the pattern $P_{2}$ is applied in columns
$B_{1}$ and $B_{3}$, while the rest of the columns use the pattern $P_{1}$.
Thus, the $(1^{6})$-coloring $c$ is given by:

$[1,2,3]_{p_{0}=0}[4,5,6]_{p_{1}=1}[1,2,3]_{p_{2}=1}[4,5,6]_{p_{3}=0}[1,2,3]_{p_{4\rightarrow
t-1}=1}[1,2,3].$

Figure 2 demonstrates how the reference point (marked in red) is shifted with
respect to the given shift sequence and how the patterns are applied.

Figure 2: $(1^{6})$-coloring of $G(k,t)$ given by patterns $[1,2,3]$ and
$[4,5,6]$, and the shift sequence $(0,1,1,0,1^{t-4}).$

To verify that $c$ is an $S$-packing coloring of $G(k,t)$ we will use
equivalent conditions, which can be derived from the above notation. Notably,
$c$ is an $S$-packing coloring of $G(k,t)$ if and only if every two vertices
with the same color are at sufficient distance, columns $B_{0}$ and $B_{t}$
obtain the same pattern and $\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\
d)$, where $d$ is the length of the pattern in $B_{0}$. The first among these
three conditions requires the following verification. Note that for every two
points $(x,y)$ and $(u,v)$ their distance in $G(k,t)$ equals
$\min\\{|x-u|+|y-v|,x+(t-u)+|(y-k)-v|\\}$. Hence, if $c(x,y)=i=c(u,v)$, then

$\min\\{|x-u|+|y-v|,x+(t-u)+|(y-k)-v|\\}>s_{i}.$

In particular, if $s_{i}=1$, then it is sufficient to verify that two vertices
with color $i$ are not adjacent in the grid. Similarly, if $s_{i}=2$, then two
vertices with color $i$ must not be adjacent in the grid, must not have a
common neighbor in the grid, and $c(1,j)=i$ implies $c(t-1,j-k)\neq i$ and
vice versa.

##  3 $S$-packing colorings of graphs $G(k,t)$

In this section, we present values of $\chi_{S}(G(k,t))$, where $3\leq k<t$
are positive integers, for all possible infinite sequences $S$ whose elements
are in $\\{1,2\\}$.

For completeness of our study of $S$-packing colorings of graphs $G(k,t)$, we
first recall the known results about the standard chromatic number. That is,
we consider the sequence $S=(1^{\infty})$. Concerning the chromatic number of
distance graphs $G(D)$, Walther proved the general bound
$\chi(G(D))\leq|D|+1$; see [1, 20]. Since $|D|=2$ in our case, we infer that
$\chi(G(k,t))\in\\{2,3\\}$ depending on whether $G(k,t)$ is bipartite or not.
Notably, for $3\leq k<t$, we derive

$\chi(G(k,t))=\left\\{\begin{array}[]{ll}2;\ k+t\textit{ even,}\\\ 3;\
k+t\textit{ odd. }\end{array}\right.$

Therefore, whenever a sequence $S$ contains three $1$s, the above result may
be applied. In the next subsection, we deal with the three remaining subcases
of sequences $S$ depending on the number of $1$s.

###  3.1 $S=(1,1,2,2,2\dots)$

###### Theorem 3.1

If $G(k,t)$ is the distance graph, where $k,t$ are coprime positive integers
such that $3\leq k<t$, and $S=(1,1,2^{\infty})$, then

$\chi_{S}(G(k,t))=\left\\{\begin{array}[]{ll}2;\ k+t\textit{ even},\\\ 4;\
k+t\textit{ odd}.\end{array}\right.$

Proof.  If $k+t$ is even, $G(k,t)$ is bipartite, hence $\chi_{S}(G(k,t))=2$.
In the rest of the proof, we assume $k+t$ is odd, which implies
$\chi_{S}(G(k,t))\geq 3$.

Suppose that $\chi_{S}(G(k,t))=3$, and let $c:V(G(k,t))\rightarrow[3]$ be a
$(1,1,2)$-coloring of $G(k,t)$. Suppose that there is a vertex $(x,y)\in
V(G(k,t))$ in column $B_{x}$, where $x\notin\\{0,t\\}$, with $c(x,y)=3$. For
this choice of $x$, the neighborhood of $(x,y)$ is the set
$N((x,y))=\\{(x,y+1),(x-1,y),(x+1,y),(x,y-1)\\}$.

Note that all vertices in $N((x,y))$ have to receive either color $1$ or $2$,
since every two vertices with color $3$ must be at the distance at least $3$.
Let $c(x,y+1)=a$, where $a\in\\{1,2\\}$, and let $\\{a,b\\}=\\{1,2\\}$. Hence,
we derive the following chain of implications:

$\begin{array}[]{lll}c(x,y+1)=a\Rightarrow c(x+1,y+1)=b\Rightarrow
c(x+1,y)=a\Rightarrow c(x+1,y-1)=b\\\ \Rightarrow c(x,y-1)=a\Rightarrow
c(x-1,y-1)=b\Rightarrow c(x-1,y)=a\Rightarrow c(x-1,y+1)=b.\\\ \end{array}$

We infer that, for every vertex $(x,y)$ with $c(x,y)=3$, all neighbors receive
the same color. (If $x\in\\{0,t\\}$, we get the same conclusion either by
checking the neighborhood of $(x,y)$ as above, or by noting that columns
$B_{0}$ and $B_{t}$ represent the same yet shifted column by which the
coloring can be reassigned so that the corresponding point with color $3$ is
in one of columns between $1$ and $t-1$.) Hence, the coloring
$c^{\prime}:V(G(k,t))\rightarrow[2]$ obtained from $c$ by recoloring every
vertex $(i,j)$ with $c(i,j)=3$ by using the color in $\\{1,2\\}$ which does
not appear in its neighborhood with respect to coloring $c$ is a
$(1,1)$-coloring of $G(k,t)$. Thus, $G(k,t)$ is bipartite, which is a
contradiction with the assumption that $k+t$ is odd. We derive that
$\chi_{S}(G(k,t))\geq 4$.

For the proof of the upper bound, we present a $(1,1,2,2)$-coloring $c$ given
by the shift sequence $(p_{i})_{i=0}^{t-1}=(0,2,0,1^{t-3})$ and two periodic
patterns $[1,2]$ and $[3,4,2,1]$ such that the pattern $[3,4,2,1]$ is used on
columns $B_{1}$ and $B_{2}$, and rest of the columns obtain pattern $[1,2]$.
That is, $c$ is defined by:

$[1,2]_{p_{0}=0}[3,4,2,1]_{p_{1}=2}[3,4,2,1]_{p_{2}=0}[1,2]_{p_{3\rightarrow
t-1}=1}[1,2].$

Note that $c$ is a $(1,1,2,2)$-coloring of $G(k,t)$ if and only if three
conditions hold: every two vertices with the same color are at sufficient
distance, columns $B_{0}$ and $B_{t}$ must obtain the same pattern and
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 2)$. The matrix below
demonstrates the presented coloring $c$ of $G(k,t)$ in the first five columns
(note that bold integers indicate the location of reference points):

⋮ ⋮ ⋮ ⋮ ⋮
---
1 3 2 1 2
2 4 1 2 1
1 2 3 1 2
2 1 4 2 1
1 3 2 1 2
2 4 1 2 1
1 2 3 1 2
2 1 4 2 1
⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at sufficient
distance and due to $t\geq 4$, columns $B_{0}$ and $B_{t}$ always use the same
pattern $[1,2]$. Thus, the first and the second condition hold. From the
following sum we obtain:

$\sum_{i=0}^{t-1}p_{i}=2+(t-3)=t-1\equiv\left\\{\begin{array}[]{ll}0\
(\mathrm{mod}\ 2);\ t\textit{ odd},\\\ 1\ (\mathrm{mod}\ 2);\ t\textit{
even}.\end{array}\right.$

Since $k$ and $t$ are of opposite parity, the condition
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 2)$ also holds.

Therefore, $\chi_{S}(G(k,t))=4$ for $k+t$ odd. $\square$

###  3.2 $S=(1,2,2,2,\dots)$

###### Theorem 3.2

If $G(k,t)$ is the distance graph, where $k,t$ are coprime positive integers
such that $3\leq k<t$, and $S=(1,2^{\infty})$, then $\chi_{S}(G(k,t))=5$.

Proof.  Due to the representation of $G(k,t)$ as the (shifted) square grid
$\\{0,1,\dots,t\\}\times\mathbb{Z}$, when determining the lower bound of
$\chi_{S}(G(k,t))$, we can use results known for the infinite grid
$\mathbb{Z}^{2}$. Goddard and Xu [15] proved that
$\chi_{S}(\mathbb{Z}^{2})=5$, hence $\chi_{S}(G(k,t))\geq 5$. (Alternatively,
the same conclusion is derived when observing a vertex in $G(k,t)$ receiving
color $1$, and noting that its neighbors must receive pairwise distinct colors
from $\\{2,\ldots,5\\}$.)

Next, we determine $(1,2^{4})$-colorings of $G(k,t)$ with respect to the
values $k$ and $t$.

  1. 1.

Let $t\geq 12$. In this case, we present six different $(1,2^{4})$-colorings
$c_{n}$, where $n\in\\{0,1,2,3,4,5\\}$, which are then applied with respect to
$k$ and $t$. The first $(1,2^{4})$-coloring $c_{0}$ is given by just one
periodic pattern $A=[1,2,3,1,4,5]$ and the shift sequence
$\mathbf{q_{0}}=(2^{t})$, while the rest of $c_{n}$ given by the shift
sequences $\mathbf{q_{n}},n\in\\{1,2,3,4,5\\}$, also use another pattern
$B=[4,1,5,2,1,3]$ in addition to pattern $A$. More precisely,

$\begin{array}[]{l}\mathbf{q_{1}}=(0,5,2^{t-2}),\\\
\mathbf{q_{2}}=((0,5)^{2},2^{t-4}),\\\ \mathbf{q_{3}}=((0,5)^{3},2^{t-6}),\\\
\mathbf{q_{4}}=((0,5)^{4},2^{t-8}),\\\ \mathbf{q_{5}}=((0,5)^{5},2^{t-10}),\\\
\end{array}$

and

$\begin{array}[]{l}c_{1}:A_{p_{0}=0}B_{p_{1}=5}A_{p_{2\rightarrow t-1}=2}A,\\\
c_{2}:A_{p_{0}=0}B_{p_{1}=5}A_{p_{2}=0}B_{p_{3}=5}A_{p_{4\rightarrow
t-1}=2}A,\\\
c_{3}:A_{p_{0}=0}B_{p_{1}=5}A_{p_{2}=0}B_{p_{3}=5}A_{p_{4}=0}B_{p_{5}=5}A_{p_{6\rightarrow
t-1}=2}A,\\\
c_{4}:A_{p_{0}=0}B_{p_{1}=5}A_{p_{2}=0}B_{p_{3}=5}A_{p_{4}=0}B_{p_{5}=5}A_{p_{6}=0}B_{p_{7}=5}A_{p_{8\rightarrow
t-1}=2}A,\\\
c_{5}:A_{p_{0}=0}B_{p_{1}=5}A_{p_{2}=0}B_{p_{3}=5}A_{p_{4}=0}B_{p_{5}=5}A_{p_{6}=0}B_{p_{7}=5}A_{p_{8}=0}B_{p_{9}=5}A_{p_{10\rightarrow
t-1}=2}A.\\\ \end{array}$

Recall that $c_{n}$ is a $(1,2^{4})$-coloring of $G(k,t)$ if and only if three
conditions hold: every two vertices with the same color are at sufficient
distance, columns $B_{0}$ and $B_{t}$ obtain the same pattern, and
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 6)$. From the definition of
colorings $c_{n}$ we immediately see that both columns $B_{0}$ and $B_{t}$
obtain the pattern $A$, hence the second condition holds.

To verify the first condition, we consider the possible pattern layouts in
three consecutive columns, $B_{i},B_{i+1},B_{i+2}$, using the patterns $A$ and
$B$. Since $B_{0}$ and $B_{t}$ represent the same, yet shifted column, the
case when $i=t-1$ is interpreted as $B_{t-1},B_{t},B_{1}$. We have five
possibilities: $AAA$, $BAA$, $ABA$, $AAB$ and $BAB$. Note that when $AAA$ is
used, in all presented sequences the two shifts are $p_{i}=2=p_{i+1}$, and it
is easy to verify that the points with the same colors are at sufficient
distances. This correspondence will be presented as:

$AAA\longleftrightarrow 2,2.$

Similarly, we have the following correspondences between pattern layouts and
shifts, which can be derived from the definitions of sequences
$\mathbf{q_{n}}$:

$\begin{array}[]{c}BAA\longleftrightarrow 5,2,\\\ ABA\longleftrightarrow
0,5,\\\ AAB\longleftrightarrow 2,0,\\\ BAB\longleftrightarrow 5,0.\\\
\end{array}$

(Note that $AAB$ appears when considering the columns $B_{t-1}$, $B_{t}$ and
$B_{1}$.) The corresponding matrices for each of the five cases are placed
below.

$\begin{array}[]{lllll}\begin{array}[]{ccc}A&A&A\\\ \vdots&\vdots&\vdots\\\
\mathbf{1}&4&3\\\ 2&5&1\\\ 3&\mathbf{1}&4\\\ 1&2&5\\\ 4&3&\mathbf{1}\\\
5&1&2\\\ 1&4&3\\\ 2&5&1\\\
\vdots&\vdots&\vdots\end{array}&\quad\begin{array}[]{ccc}A&A&B\\\
\vdots&\vdots&\vdots\\\ \mathbf{1}&4&1\\\ 2&5&3\\\ 3&\mathbf{1}&\mathbf{4}\\\
1&2&1\\\ 4&3&5\\\ 5&1&2\\\ 1&4&1\\\ 2&5&3\\\
\vdots&\vdots&\vdots\end{array}&\quad\begin{array}[]{ccc}A&B&A\\\
\vdots&\vdots&\vdots\\\ \mathbf{1}&\mathbf{4}&2\\\ 2&1&3\\\ 3&5&1\\\ 1&2&4\\\
4&1&5\\\ 5&3&\mathbf{1}\\\ 1&4&2\\\ 2&1&3\\\
\vdots&\vdots&\vdots\end{array}&\quad\begin{array}[]{ccc}B&A&A\\\
\vdots&\vdots&\vdots\\\ \mathbf{4}&2&5\\\ 1&3&1\\\ 5&1&2\\\ 2&4&3\\\ 1&5&1\\\
3&\mathbf{1}&4\\\ 4&2&5\\\ 1&3&\mathbf{1}\\\
\vdots&\vdots&\vdots\end{array}&\quad\begin{array}[]{ccc}B&A&B\\\
\vdots&\vdots&\vdots\\\ \mathbf{4}&2&1\\\ 1&3&5\\\ 5&1&2\\\ 2&4&1\\\ 1&5&3\\\
3&\mathbf{1}&\mathbf{4}\\\ 4&2&1\\\ 1&3&5\\\
\vdots&\vdots&\vdots\end{array}\end{array}$

Again, in all cases it is easy to verify that the points with the same colors
are at sufficient distances. In this way, the first condition is also
verified.

What remains is to use the third condition $\sum_{i=0}^{t-1}p_{i}\equiv k\
(\mathrm{mod}\ 6)$ to determine which sequence $\mathbf{q_{n}}$ is suitable
for $G(k,t)$ with respect to values $k,t$. Let $t\equiv\ell\ (\mathrm{mod}\
6)$ where $\ell\in\\{0,1,2,3,4,5\\}$. For each $\mathbf{q_{n}}$ we calculate
the value of $\sum_{i=0}^{t-1}p_{i}$:

$\mathbf{q_{n}}:\sum_{i=0}^{t-1}p_{i}=5n+2(t-2n)=2t+n.$

Due to appropriate numbering of sequences we obtain
$\sum_{i=0}^{t-1}p_{i}\equiv(2\ell+n)\ (\mathrm{mod}\ 6)$ for each
$\mathbf{q_{n}}$. In order to determine which $c_{n}$ gives us the
$(1,2^{4})$-coloring of $G(k,t)$ for fixed values $k,t$, let $k\equiv m\
(\mathrm{mod}\ 6)$ where $m\in\\{0,1,2,3,4,5\\}$. Using the condition
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 6)$ we derive:

$\begin{array}[]{c}(2\ell+n)\ (\mathrm{mod}\
6)=m\,\Longrightarrow\,n=(m-2\ell)\ (\mathrm{mod}\ 6).\end{array}$

Thus, if $k\equiv m\ (\mathrm{mod}\ 6)$ and $t\equiv\ell\ (\mathrm{mod}\ 6)$
then the $(1,2^{4})$-coloring of $G(k,t)$ is given by $c_{n}$ using the
sequence $\mathbf{q_{n}}$ such that $n=(m-2\ell)\ (\mathrm{mod}\ 6)$.

  2. 2.

Let $t=11$, hence $k\in\\{3,4,5,6,7,8,9,10\\}$. For $G(3,11)$ and $G(8,11)$ we
present a $(1,2^{4})$-coloring given by the periodic pattern $[1,2,3,4,5]$ and
the shift sequence $(p_{i})_{i=0}^{10}=(3^{11})$. The matrix below
demonstrates the presented $(1,2^{4})$-coloring of these graphs:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 3 5 2 4 1 3 5 2 4 1 3
2 4 1 3 5 2 4 1 3 5 2 4
3 5 2 4 1 3 5 2 4 1 3 5
4 1 3 5 2 4 1 3 5 2 4 1
5 2 4 1 3 5 2 4 1 3 5 2
1 3 5 2 4 1 3 5 2 4 1 3
2 4 1 3 5 2 4 1 3 5 2 4
3 5 2 4 1 3 5 2 4 1 3 5
4 1 3 5 2 4 1 3 5 2 4 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{10}p_{i}\equiv 3\ (\mathrm{mod}\ 5)$
holds for both graphs.

For $G(4,11)$, $G(5,11)$, $G(6,11)$, $G(7,11)$, $G(9,11)$ and $G(10,11)$ we
use $(1,2^{4})$-colorings $c_{0}$, $c_{1}$, $c_{2}$, $c_{3}$, $c_{5}$ and
$c_{0}$ from case $1$, respectively.

  3. 3.

Let $t=10$, hence $k\in\\{3,7,9\\}$. For both $G(3,10)$ and $G(9,10)$ we use
$(1,2^{4})$-coloring $c_{1}$ from case $1$.

For $G(7,10)$ we present a $(1,2^{4})$-coloring given by two periodic patterns
$C=[1,2,1,3,1,4,1,5]$ and $D=[4,3,5,4,2,5,3,2]$, and the shift sequence
$(p_{i})_{i=0}^{9}=((0,1)^{3},3^{4})$ such that:

$C_{p_{0}=0}D_{p_{1}=1}C_{p_{2}=0}D_{p_{3}=1}C_{p_{4}=0}D_{p_{5}=1}C_{p_{6\rightarrow
9}=3}C.$

The matrix below demonstrates the presented $(1,2^{4})$-coloring of this
graph:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 4 5 2 1 3 4 1 5 1 2
2 3 1 4 5 2 1 3 1 4 1
1 5 2 3 1 4 5 1 2 1 3
3 4 1 5 2 3 1 4 1 5 1
1 2 3 4 1 5 2 1 3 1 4
4 5 1 2 3 4 1 5 1 2 1
1 3 4 5 1 2 3 1 4 1 5
5 2 1 3 4 5 1 2 1 3 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{9}p_{i}\equiv 7\ (\mathrm{mod}\ 8)$
holds.

  4. 4.

Let $t=9$, hence $k\in\\{4,5,7,8\\}$. For $G(4,9)$ we present a
$(1,2^{4})$-coloring given by two periodic patterns $C=[1,2,1,3,1,4,1,5]$ and
$D=[4,3,5,4,2,5,3,2]$, and the shift sequence
$(p_{i})_{i=0}^{8}=((0,1)^{3},3^{3})$ such that:

$C_{p_{0}=0}D_{p_{1}=1}C_{p_{2}=0}D_{p_{3}=1}C_{p_{4}=0}D_{p_{5}=1}C_{p_{6\rightarrow
8}=3}C.$

The matrix below demonstrates the presented $(1,2^{4})$-coloring of this
graph:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 4 5 2 1 3 4 1 5 1
2 3 1 4 5 2 1 3 1 4
1 5 2 3 1 4 5 1 2 1
3 4 1 5 2 3 1 4 1 5
1 2 3 4 1 5 2 1 3 1
4 5 1 2 3 4 1 5 1 2
1 3 4 5 1 2 3 1 4 1
5 2 1 3 4 5 1 2 1 3
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{8}p_{i}\equiv 4\ (\mathrm{mod}\ 8)$
holds.

For $G(5,9)$ we present a $(1,2^{4})$-coloring given by the periodic pattern
$C=[1,2,1,3,1,4,1,5]$ and the shift sequence $(p_{i})_{i=0}^{8}=(5^{9})$.
Again, the matrix below demonstrates the presented $(1,2^{4})$-coloring of
this graph:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 3 1 2 1 5 1 4 1 3
2 1 5 1 4 1 3 1 2 1
1 4 1 3 1 2 1 5 1 4
3 1 2 1 5 1 4 1 3 1
1 5 1 4 1 3 1 2 1 5
4 1 3 1 2 1 5 1 4 1
1 2 1 5 1 4 1 3 1 2
5 1 4 1 3 1 2 1 5 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{8}p_{i}\equiv 5\ (\mathrm{mod}\ 8)$
holds.

For $G(7,9)$ and $G(8,9)$ we use $(1,2^{4})$-colorings $c_{1}$ and $c_{2}$
from case $1$, respectively.

  5. 5.

Let $t=8$, hence $k\in\\{3,5,7\\}$. For $G(3,8)$ we present a
$(1,2^{4})$-coloring given by two periodic patterns $C=[1,2,1,3,1,4,1,5]$ and
$D=[4,3,5,4,2,5,3,2]$, and the shift sequence $(p_{i})_{i=0}^{7}=(0,1,3^{6})$
such that:

$C_{p_{0}=0}D_{p_{1}=1}C_{p_{2\rightarrow 7}=3}C.$

The matrix below demonstrates the presented $(1,2^{4})$-coloring of this
graph:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 4 5 1 2 1 3 1 4
2 3 1 4 1 5 1 2 1
1 5 2 1 3 1 4 1 5
3 4 1 5 1 2 1 3 1
1 2 3 1 4 1 5 1 2
4 5 1 2 1 3 1 4 1
1 3 4 1 5 1 2 1 3
5 2 1 3 1 4 1 5 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{7}p_{i}\equiv 3\ (\mathrm{mod}\ 8)$
holds.

For $G(5,8)$ and $G(7,8)$ we use $(1,2^{4})$-colorings $c_{1}$ and $c_{3}$
from case $1$, respectively.

  6. 6.

Let $t=7$, hence $k\in\\{3,4,5,6\\}$. For $G(3,7)$, $G(4,7)$ and $G(5,7)$ we
use $(1,2^{4})$-colorings $c_{1}$, $c_{2}$ and $c_{3}$ from case $1$,
respectively.

For $G(6,7)$ we present a $(1,2^{4})$-coloring given by the periodic pattern
$[1,2,3,4,5]$ and the shift sequence $(p_{i})_{i=0}^{6}=(3^{7})$. The matrix
below demonstrates the presented $(1,2^{4})$-coloring of this graph:

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 3 5 2 4 1 3 5
2 4 1 3 5 2 4 1
3 5 2 4 1 3 5 2
4 1 3 5 2 4 1 3
5 2 4 1 3 5 2 4
1 3 5 2 4 1 3 5
2 4 1 3 5 2 4 1
3 5 2 4 1 3 5 2
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and $\sum_{i=0}^{6}p_{i}\equiv 1\ (\mathrm{mod}\ 5)$, hence the
condition $\sum_{i=0}^{6}p_{i}\equiv k\ (\mathrm{mod}\ 5)$ holds.

  7. 7.

Let $t=6$, hence $k=5$. For $G(5,6)$ we present an $(1,2^{4})$-coloring given
by two periodic patterns $C=[1,2,1,3,1,4,1,5]$ and $D=[4,3,5,4,2,5,3,2]$, and
the shift sequence $(p_{i})_{i=0}^{5}=(0,1,3^{4})$ such that:

$C_{p_{0}=0}D_{p_{1}=1}C_{p_{2\rightarrow 5}=3}C.$

The matrix below demonstrates the presented $(1,2^{4})$-coloring of this
graph.

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 4 5 1 2 1 3
2 3 1 4 1 5 1
1 5 2 1 3 1 4
3 4 1 5 1 2 1
1 2 3 1 4 1 5
4 5 1 2 1 3 1
1 3 4 1 5 1 2
5 2 1 3 1 4 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{4}p_{i}\equiv 5\ (\mathrm{mod}\ 8)$
holds.

  8. 8.

Let $t=5$, hence $k\in\\{3,4\\}$. For $G(3,5)$ we present an
$(1,2^{4})$-coloring given by the periodic pattern
$[(1,2,1,3)^{2},(1,4,1,5)^{2}]$ and the shift sequence
$(p_{i})_{i=0}^{4}=(7^{5})$. The matrix below demonstrates the presented
$(1,2^{4})$-coloring of this graph.

⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 4 1 5 1 4
2 1 3 1 2 1
1 5 1 4 1 5
3 1 2 1 3 1
1 4 1 5 1 2
2 1 3 1 4 1
1 5 1 2 1 3
3 1 4 1 5 1
1 2 1 3 1 2
4 1 5 1 4 1
1 3 1 2 1 3
5 1 4 1 5 1
1 2 1 3 1 4
4 1 5 1 2 1
1 3 1 4 1 5
5 1 2 1 3 1
⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{4}p_{i}\equiv 3\ (\mathrm{mod}\ 16)$
holds.

For $G(4,5)$ we use coloring $c_{0}$ from case $1$.

  9. 9.

Let $t=4$, hence $k=3$. For $G(3,4)$ we use coloring $c_{1}$ from case $1$.

After verifying all the possible cases for $k$ and $t$, we conclude that
$\chi_{S}(G(k,t))=5$, and the proof is complete. $\square$

###  3.3 $S=(2,2,2,\dots)$

###### Lemma 3.3

If $G(k,t)$ is the distance graph, where $k,t$ are coprime positive integers
such that $3\leq k<t$, and $S=(2^{\infty})$, then $\chi_{S}(G(k,t))\leq 6$.

Proof.  We determine $(2^{6})$-colorings of $G(k,t)$ with respect to the
values of $k,t$.

  1. 1.

Let $t\geq 6$. In this case, we present six different $(2^{6})$-colorings
$c_{n}$, where $n\in\\{0,1,2,3,4,5\\}$, which we apply with respect to $k$ and
$t$. Each $c_{n}$ is given by the periodic pattern $[1,2,3,4,5,6]$ and a shift
sequence $\mathbf{q_{n}}=(p_{i})_{i=0}^{t-1}$ such that:

$\begin{array}[]{l}\mathbf{q_{0}}=(2^{t}),\\\ \mathbf{q_{1}}=(3,2^{t-1}),\\\
\mathbf{q_{2}}=(3,2,3,2^{t-3}),\\\ \mathbf{q_{3}}=(3,2,3,2,3,2^{t-5}),\\\
\mathbf{q_{4}}=(3,4,3,2^{t-3}),\\\ \mathbf{q_{5}}=(3,4,3,2,3,2^{t-5}).\\\
\end{array}$

Since the colorings are given by the unique pattern, $c_{n}$ is a
$(2^{6})$-coloring of $G(k,t)$ if and only if every two vertices with the same
color are at the distance at least $3$ and $\sum_{i=0}^{t-1}p_{i}\equiv k\
(\mathrm{mod}\ 6)$. To verify the first condition, we observe a coloring
$c:V(\mathbb{Z}^{2})\rightarrow[6]$ of the infinite grid $\mathbb{Z}^{2}$
using the pattern $[1,2,3,4,5,6]$. Since $c$ is $(2^{6})$-coloring, the
following must hold for each $i,j\in\mathbb{Z}$:

$\begin{array}[]{l}c(i,j)\neq c(i+1,j),\\\ c(i,j)\neq c(i+1,j\pm 1),\\\
c(i,j)\neq c(i+2,j).\end{array}$

Thus, the periodic pattern $[1,2,3,4,5,6]$ in column $B_{i+1}$ can be shifted
by $2,3$ or $4$ with respect to the column $B_{i}$. In addition, the pattern
in column $B_{i+2}$ can be shifted by $1,2,3,4$ or $5$ with respect to the
column $B_{i}$. Thus, for the graph $G(k,t)$ we obtain $p_{i}\in\\{2,3,4\\}$,
and

$\begin{array}[]{ccc}p_{i}=2\Rightarrow p_{i+1}\neq 4,\\\ p_{i}=3\Rightarrow
p_{i+1}\neq 3,\\\ p_{i}=4\Rightarrow p_{i+1}\neq 2,\\\ \end{array}$

holds for all $i\in\\{0,\dots,t-2\\}$, and, furthermore,

$\begin{array}[]{ccc}p_{0}=2\Rightarrow p_{t-1}\neq 4,\\\ p_{0}=3\Rightarrow
p_{t-1}\neq 3,\\\ p_{0}=4\Rightarrow p_{t-1}\neq 2,\\\ \end{array}$

because $B_{0}$ and $B_{t}$ represents the same yet shifted column.

Next, we use the second condition $\sum_{i=0}^{t-1}p_{i}\equiv k\
(\mathrm{mod}\ 6)$ to determine which sequence $\mathbf{q_{n}}$ is suitable
for $G(k,t)$ with respect to values $k,t$. Let $t\equiv\ell\ (\mathrm{mod}\
6)$ where $\ell\in\\{0,1,2,3,4,5\\}$. For each $\mathbf{q_{n}}$ we calculate
the value of $\sum_{i=0}^{t-1}p_{i}$:

$\begin{array}[]{l}\mathbf{q_{0}}:\sum_{i=0}^{t-1}p_{i}=2t\equiv 2\ell\
(\mathrm{mod}\ 6),\\\ \\\
\mathbf{q_{1}}:\sum_{i=0}^{t-1}p_{i}=3+2(t-1)=2t+1\equiv(2\ell+1)\
(\mathrm{mod}\ 6),\\\ \\\
\mathbf{q_{2}}:\sum_{i=0}^{t-1}p_{i}=3+2+3+2(t-3)=2t+2\equiv(2\ell+2)\
(\mathrm{mod}\ 6),\\\ \\\
\mathbf{q_{3}}:\sum_{i=0}^{t-1}p_{i}=3+2+3+2+3+2(t-5)=2t+3\equiv(2\ell+3)\
(\mathrm{mod}\ 6),\\\ \\\
\mathbf{q_{4}}:\sum_{i=0}^{t-1}p_{i}=3+4+3+2(t-3)=2t+4\equiv(2\ell+4)\
(\mathrm{mod}\ 6),\\\ \\\
\mathbf{q_{5}}:\sum_{i=0}^{t-1}p_{i}=3+4+3+2+3+2(t-5)=2t+5\equiv(2\ell+5)\
(\mathrm{mod}\ 6).\\\ \end{array}$

Thus, for $\mathbf{q_{n}}$ we obtain $\sum_{i=0}^{t-1}p_{i}\equiv(2\ell+n)\
(\mathrm{mod}\ 6)$. In order to determine which $c_{n}$ gives us the
$(2^{6})$-coloring of $G(k,t)$ for fixed values $k,t$, let $k\equiv m\
(\mathrm{mod}\ 6)$ where $m\in\\{0,1,2,3,4,5\\}$. From the condition
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 6)$ we obtain:

$\begin{array}[]{c}(2\ell+n)\ (\mathrm{mod}\
6)=m\,\Longrightarrow\,n=(m-2\ell)\ (\mathrm{mod}\ 6),\end{array}$

hence if $k\equiv m\ (\mathrm{mod}\ 6)$ and $t\equiv\ell\ (\mathrm{mod}\ 6)$
then the $(2^{6})$-coloring of $G(k,t)$ is given by $c_{n}$ using the sequence
$\mathbf{q_{n}}$ such that $n=(m-2\ell)\ (\mathrm{mod}\ 6)$.

  2. 2.

Let $t=5$, hence $k\in\\{3,4\\}$. For $G(3,5)$ we present a $(2^{6})$-coloring
given by the periodic pattern $[1,2,3,4,5,1,6,3,2,5,4,6]$ and the shift
sequence $(p_{i})_{i=0}^{4}=(3^{5})$. The matrix below demonstrates the
presented $(2^{6})$-coloring of this graph.

⋮ ⋮ ⋮ ⋮ ⋮ ⋮
---
1 5 6 4 1 5
2 4 3 5 2 4
3 6 2 1 3 6
4 1 5 6 4 1
5 2 4 3 5 2
1 3 6 2 1 3
6 4 1 5 6 4
3 5 2 4 3 5
2 1 3 6 2 1
5 6 4 1 5 6
4 3 5 2 4 3
6 2 1 3 6 2
⋮ ⋮ ⋮ ⋮ ⋮ ⋮

It is easy to verify that all vertices with the same color are at a sufficient
distance and the condition $\sum_{i=0}^{4}p_{i}\equiv 3\ (\mathrm{mod}\ 12)$
holds.

For $G(4,5)$ we use the coloring $c_{0}$ from case $1$.

  3. 3.

Let $t=4$, hence $k=3$. For $G(3,4)$ we use the coloring $c_{1}$ from case
$1$.

By considering all cases of $k$ and $t$, we obtain $\chi_{S}(G(k,t))\leq 6$,
as desired. $\square$

###### Theorem 3.4

If $G(k,t)$ is the distance graph, where $k,t$ are coprime positive integers
such that $3\leq k<t$, and $S=(2^{\infty})$, then

$\chi_{S}(G(k,t))=\left\\{\begin{array}[]{ll}5;\ (t\equiv 1,4\ (\mathrm{mod}\
5)\textit{ and }k\equiv 2,3\ (\mathrm{mod}\ 5))\\\ \ \ \ \textit{ or }(t\equiv
2,3\ (\mathrm{mod}\ 5)\textit{ and }k\equiv 1,4\ (\mathrm{mod}\ 5)),\\\ \\\
6;\ \textit{ otherwise}.\end{array}\right.$

Proof.  Consider the infinite grid $\mathbb{Z}^{2}$. Goddard and Xu [15]
proved that $\chi_{S}(\mathbb{Z}^{2})=5$, hence $\chi_{S}(G(k,t))\geq 5$. Let
$c:V(\mathbb{Z}^{2})\rightarrow[5]$ be a $(2^{5})$-coloring of
$\mathbb{Z}^{2}$. Consider a vertex $(x,y)\in V(\mathbb{Z}^{2})$ with its
neighborhood $N((x,y))=\\{(x,y+1),(x-1,y),(x+1,y),(x,y-1)\\}$. Each of these
$5$ vertices must receive a different color since they are at the distance at
most $2$. Without loss of generality, let

$\begin{array}[]{ccc}c(x,y+1)=1,\ c(x-1,y)=2,\ c(x,y)=3,\ c(x+1,y)=4,\
c(x,y-1)=5.\end{array}$

Thus, the vertex $(x+1,y+1)$ can obtain either color $2$ or $5$ with respect
to the coloring $c$. We consider both options.

  1. 1.

Let $c(x+1,y+1)=2$. Due to the assignment of colors to vertices in $N([x,y])$
and $(x+1,y+1)$, we derive the following chain of implications:

$\begin{array}[]{lll}c(x+1,y+1)=2\Rightarrow c(x+1,y-1)=1\Rightarrow
c(x-1,y-1)=4\\\ \Rightarrow c(x,y-2)=2\Rightarrow c(x+1,y-2)=3\Rightarrow
c(x-1,y-2)=1\\\ \Rightarrow c(x,y-3)=4\Rightarrow c(x+1,y-3)=5\Rightarrow
c(x-1,y-3)=3\\\ \Rightarrow c(x,y-4)=1\Rightarrow\dots\end{array}$

Thus, the coloring $c$ forces for each column $B_{i}$ the periodic pattern
$[1,3,5,2,4]$. Moreover, $c(i,j)=c(i+1,j-2)$ for all $i,j\in\mathbb{Z}$.

  2. 2.

Let $c(x+1,y+1)=5$. Similarly:

$\begin{array}[]{lll}c(x+1,y+1)=5\Rightarrow c(x-1,y+1)=4\Rightarrow
c(x-1,y-1)=1\\\ \Rightarrow c(x+1,y-1)=2\Rightarrow c(x,y-2)=4\Rightarrow
c(x-1,y-2)=3\\\ \Rightarrow c(x+1,y-2)=1\Rightarrow c(x,y-3)=2\Rightarrow
c(x-1,y-3)=5\\\ \Rightarrow c(x+1,y-3)=3\Rightarrow
c(x,y-4)=1\Rightarrow\dots\end{array}$

In this case, the coloring $c$ forces for each column $B_{i}$ the periodic
pattern $[1,3,5,4,2]$ and $c(i,j)=c(i+1,j-3)$ for all $i,j\in\mathbb{Z}$.

We have shown that (up to permutation of colors) there exist only two
$2^{5}$-packing colorings of the square lattice $\mathbb{Z}^{2}$. We now apply
these two colorings to $G(k,t)$.

  1. 1.

Consider the coloring given by the periodic pattern $[1,3,5,2,4]$ with
relation $c(i,j)=c(i+1,j-2)$. Hence, this coloring enforces the constant shift
sequence $(p_{i})_{i=0}^{t-1}$ where $p_{i}=2$ for all
$i\in\\{0,\dots,t-1\\}$. Note that if $c$ is a $(2^{5})$-coloring of $G(k,t)$,
then $\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 5)$. What remains is to
determine for which values of $k,t$ this condition holds.

Let $t\equiv\ell\ (\mathrm{mod}\ 5)$ where $\ell\in\\{0,1,2,3,4\\}.$ From the
following sum we derive:

$\sum_{i=0}^{t-1}p_{i}=\sum_{i=0}^{t-1}2=2t\equiv 2\ell\ (\mathrm{mod}\ 5),$

hence $k\equiv 2\ell\ (\mathrm{mod}\ 5)$. Note that for $\ell=0$ we obtain
$t,k\equiv 0\ (\mathrm{mod}\ 5)$ which is a contradiction with the proposition
$\gcd(k,t)=1$ and thus it is sufficient to consider $\ell\in\\{1,2,3,4\\}$.

  2. 2.

Consider the coloring given by the periodic pattern $[1,3,5,4,2]$ with
relation $c(i,j)=c(i+1,j-3)$, which is equivalent to the constant shift
sequence $(p_{i})_{i=0}^{t-1}$ where $p_{i}=3$ for all
$i\in\\{0,\dots,t-1\\}$. We again determine for which $k,t$ the condition
$\sum_{i=0}^{t-1}p_{i}\equiv k\ (\mathrm{mod}\ 5)$ holds.

Let $t\equiv\ell\ (\mathrm{mod}\ 5)$ where $\ell\in\\{0,1,2,3,4\\}.$ Similarly
to the previous case:

$\sum_{i=0}^{t-1}p_{i}=\sum_{i=0}^{t-1}3=3t\equiv 3\ell\ (\mathrm{mod}\ 5),$

hence $k\equiv 3\ell\ (\mathrm{mod}\ 5)$. For $\ell=0$ we obtain $t,k\equiv 0\
(\mathrm{mod}\ 5)$, contradicting $\gcd(k,t)=1$ again, thus we consider
$\ell\in\\{1,2,3,4\\}$.

By calculating the values of $k,t$ for the given $\ell$, we obtain that
$\chi_{S}(G(k,t))=5$ if and only if either $(t\equiv 1,4\ (\mathrm{mod}\ 5)$
and $k\equiv 2,3\ (\mathrm{mod}\ 5))$ or $(t\equiv 2,3\ (\mathrm{mod}\ 5)$ and
$k\equiv 1,4\ (\mathrm{mod}\ 5))$. From Lemma 3.3 we derive
$\chi_{S}(G(k,t))=6$ for the remaining values of $k,t$. $\square$

##  4 Concluding remarks

These results presented in this paper complement previously known results from
[1, 2, 16, 20] on $S$-packing colorings of connected distance graphs $G(k,t)$,
which satisfy the property that each integer in a sequence $S$ belongs to
$\\{1,2\\}$. Note that such $S$-packing colorings are classical colorings
($S=(1^{\infty})$), $2$-distance colorings ($S=(2^{\infty})$) and $S$-packing
colorings which lie between these two.

The amalgamation of the results presented herein with those established
earlier gives us the $S$-packing chromatic numbers of all connected distance
graphs $G(k,t)$, where $k\geq 1$ and $t>k$ are coprime. With respect to the
sequence $S$ we summarize all of these results as follows.

  1. 1.

$S=(1^{\infty})$.

$\chi(G(k,t))=\left\\{\begin{array}[]{ll}2;\ k+t\textit{ even,}\\\ 3;\
k+t\textit{ odd. }\end{array}\right.$

  2. 2.

$S=(1,1,2^{\infty})$.

$\chi_{S}(G(k,t))=\left\\{\begin{array}[]{ll}2;\ k+t\textit{ even},\\\ 4;\
k+t\textit{ odd}.\end{array}\right.$

  3. 3.

$S=(1,2^{\infty})$.

$\chi_{S}(G(k,t))=\left\\{\begin{array}[]{ll}5;\ k\neq 2\textit{ or }t\neq
3,\\\ 6;\ \textit{otherwise}.\end{array}\right.$

  4. 4.

$S=(2^{\infty})$.

$\chi_{2}(G(k,t))=\left\\{\begin{array}[]{ll}5;\ (t\equiv 1,4\ (\mathrm{mod}\
5)\textit{ and }k\equiv 2,3\ (\mathrm{mod}\ 5))\\\ \ \ \ \textit{ or }(t\equiv
2,3\ (\mathrm{mod}\ 5)\textit{ and }k\equiv 1,4\ (\mathrm{mod}\ 5)),\\\ \\\
7;\ k=2\textit{ and }t=3,\\\ 6;\ \textit{ otherwise}.\end{array}\right.$

As for other sequences $S$ that contain elements greater than $2$,
investigations of $\chi_{S}(G(k,t))$ are far from complete. For the sequence
$S=(d^{\infty})$, $d\geq 3$, lower and upper bounds are known for
$\chi_{S}(G(k,t))$ [2], which in some cases culminate to exact results. For
instance, if $t\geq 5$ is an odd integer and $d\geq t-3$, then
$\chi_{d}(G(k,t))=1+t\cdot\left(d-\frac{t-3}{2}\right)$. Similarly, lower and
upper bounds are known for the sequence $S=(1,2,3,\ldots)$ which corresponds
to the standard packing coloring. This results focus on the distance graph
$G(1,t)$ [10, 19]. Additionally, exact results exist for certain sporadic
sequences $S$, which came into fruition while studying the $S$-packing
chromatic numbers of $G(k,t)$ for sequences that contain only elements from
$\\{1,2\\}$. The $S$-packing coloring of $G(1,t)$ provided in [16] partitions
the color classes in such a way that the vertices of some color classes are
farther apart than they need to be. As a consequence, this gives results for
the sequences $S$ with larger elements.

There remains ample scope for research concerning the $S$-packing chromatic
number of distance graphs $G(k,t)$. In addition, for distance graphs $G(D)$,
where $|D|\geq 3$, only a few results on their $S$-packing colorings are
known.

## Acknowledgements

B.B., J.F. and M.J. acknowledge the financial support of the Slovenian
Research and Innovation Agency (research core funding No. P1-0297 and projects
N1-0285, J1-3002, and J1-4008).

## References

  * [1] B. Benmedjdoub, I. Bouchemakh, É. Sopena, 2-distance colorings of integer distance graphs, Discuss. Math. Graph Theory 39 (2019) 589-603.
  * [2] B. Brešar, J. Ferme, K. Kamenická, $S$-packing colorings of distance graphs $G(\mathbb{Z},\\{2,t\\})$, Discrete Appl. Math. 298 (2021) 143–154.
  * [3] B. Brešar, J. Ferme, S. Klavžar and D.F. Rall, A survey on packing colorings, Discuss. Math. Graph Theory 40 (2020) 923–970.
  * [4] B. Brešar, S. Klavžar, D.F. Rall, On the packing chromatic number of Cartesian products, hexagonal lattice, and trees, Discrete Appl. Math. 155 (2007) 2303–2311.
  * [5] B. Brešar, S. Klavžar, D.F. Rall and K. Wash, Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph, Aequationes Math. 91 (2017) 169–184.
  * [6] J. Chen, G. Chang, K. Huang, Integral distance graphs, J. Graph Theory 25 (1997) 287-294.
  * [7] D. Der-Fen Liu, A. Sutedja, Chromatic number of distance graphs generated by the sets $\\{2,3,x,y\\}$, Journal Combin. Optim. 25 (2013) 680-693.
  * [8] W.A. Deuber, X. Zhu, The chromatic numbers of distance graphs, Discrete Math. 165/166 (1997) 195-204.
  * [9] R.B. Eggleton, P. Erdős, D.K. Skilton, Colouring the real line, J. Combin. Theory ser. B 39 (1985) 86-100.
  * [10] J. Ekstein, P. Holub and B. Lidický, Packing chromatic number of distance graphs, Discrete Appl. Math. 160 (2012) 518–524.
  * [11] J. Ekstein, P. Holub and O. Togni, The packing coloring of distance graphs $D(k,t)$, Discrete Appl. Math. 167 (2014) 100–106.
  * [12] N. Gastineau and O. Togni, $S$-packing colorings of cubic graphs, Discrete Math. 339 (2016) 2461–2470.
  * [13] W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, J.M. Harris and D.F. Rall, Broadcast chromatic numbers of graphs, Ars Combin. 86 (2008) 33–49.
  * [14] W. Goddard and H. Xu, The $S$-packing chromatic number of a graph, Discuss. Math. Graph Theory 32 (2012) 795–806.
  * [15] W. Goddard and H. Xu, A note on $S$-packing colorings of lattices, Discrete Appl. Math. 166 (2014) 255–262.
  * [16] P. Holub, J. Hofman, On S-packing colourings of distance graphs $D(1,t)$ and $D(1,2,t)$, Appl. Math. Comput. 447 (2023) Paper No. 127855.
  * [17] F. Kramer and H. Kramer, A survey on the distance-colouring of graphs, Discrete Math. 308 (2008) 422–426.
  * [18] R. Liu, X. Liu, M. Rolek and G. Yu, Packing $(1,1,2,2)$-coloring of some subcubic graphs, Discrete Appl. Math. 283 (2020) 626-630.
  * [19] O. Togni, On packing colorings of distance graphs, Discrete Appl. Math. 167 (2014) 280–289.
  * [20] H. Walther, Über eine spezielle Klasse unendlicher Graphen, in: Graphentheorie, K. Wagner and R. Bodendiek (Ed(s)), (Bibl. Inst., Mannheim, 1990) 2, 268–295.