††thanks: These authors contributed equally.††thanks: These authors
contributed equally.††thanks: Corresponding author

#  Spontaneous frequency shift and phase delay of coupled terahertz radiations
mediated by the Josephson plasmon in a cuprate superconductor

Ryota Kobayashi  Ken Hayama  Shuma Fujita  Department of Electronic Science
and Engineering, Kyoto University, Kyotodaigaku Katsura, Nishikyo, Kyoto
615-8510, Japan  Manabu Tsujimoto  Research Center for Emerging Computing
Technologies, National Institute of Advanced Industrial Science and Technology
(AIST), Central2, 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan.  Faculty of
Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba,
Ibaraki 305-8573, Japan  Itsuhiro Kakeya<EMAIL_ADDRESS>Department
of Electronic Science and Engineering, Kyoto University, Kyotodaigaku Katsura,
Nishikyo, Kyoto 615-8510, Japan

###### Abstract

We examine coupling interactions used to synchronize macroscopic Josephson
oscillations induced in intrinsic Josephson junction (IJJ) mesa stacks made of
a Bi2212 single crystal. Synchronized radiations of terahertz electromagnetic
(EM) waves are detected under common voltage and current bias operations of
two connected mesas with close individual radiation frequencies, while
uncoupled and bimodal radiations are frequently observed in two mesas with
different individual radiation frequencies. Detailed observations of the
polarizations of the EM waves emitted when two mesas are biased in parallel or
series allow us to reveal the coupling matrix components, which include ratios
of synchronized IJJs in the mesas and phase delay between the macroscopic
Josephson oscillations. A frequency evolution of the phase delay implies that
the coupling between the Josephson oscillations is mediated by the small
amplitude Josephson plasmon inside the superconducting substrate. This finding
stimulates systematic survey on polarization of EM wave emitted from
synchronized multiple mesa devices in order to realize powerful terahertz
emissions from superconductors.

##  I Introduction

Various phenomena attributed to couplings between Josephson junctions have
been observed in cuprate superconductor single crystals, in which a strong
modulation of order parameter along the $c$-axis attributed to the layered
crystal structure exists [1]. In such systems, known as intrinsic Josephson
junctions (IJJs), the excitation of a Josephson plasma wave along the
$ab$-plane [2, 3] induces macroscopic current oscillations on the crystal
surface owing to the interplay with the alternating current (AC) Josephson
effect and their synchronous oscillations accompanied by the inductive and the
capacitive couplings among stacked IJJs [4, 5, 6, 7], which combine to cause
electromagnetic (EM) radiation into space [8, 9, 10]. Intensive research has
been carried out on Josephson plasma emission (JPE) using mesa-structured
devices formed from Bi2Sr2CaCu2O8+δ (Bi2212) single crystals [11, 12, 13, 14,
15, 16, 17, 18, 19, 20], and further research on the possibility of a novel
terahertz (THz) source have been reported during this decade [21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33]. For the implementation of this JPE as a
practical terahertz source, it is indispensable to manage the synchronous
oscillations resulting in higher radiation power and controlled performances.
However, the microscopic interaction between thousands of IJJs included in the
mesa-devices have not been fully revealed partly because of the lack of
experimental results in a single-layer IJJ with the similar $ab$-plane
geometry to existing JPE devices.

One of the straightforward strategies to increase the radiation intensity is
to synchronize multiple mesas to emit radiation. As an initial attempt, when
two mesas were connected in series, the coupled emission intensity was found
to be higher than the sum of the individual intensities[34]. Next, for a
device with up to three mesas connected in parallel, an intensity of 0.61 mW
was achieved, which is roughly proportional to the square of the number of
mesas to be synchronized, and it was concluded that the emission of the
electric wave is a synchronous coherent radiation [35].

Very recently, the authors proposed a method for analyzing the synchronous
phenomena with bases as polarizations of singly biased radiations and a
coupling matrix between the Josephson plasma waves excited in the mesas from
precise observations of the polarization of the radiated EM wave [36, 37].
This method allows us to determine the contribution of each mesa to the
synchronized emission under biasing multiple mesas. It is crucial to compare
the emission features for parallel and series connections of a pair of mesas
because the coupling matrix can be modified with the identical bases. However,
there has been no report of investigations on the differences between them in
an identical device. Regarding a radiating mesa in which a thousand stacked
IJJs oscillate coherently as a macroscopic Josephson junction [38], the common
voltage bias (CVB) produced by the parallel connection should induce
synchronous oscillations between the macroscopic Josephson junctions, whereas
the common current bias (CCB) produced by the series connection give insights
for the interplay between a few stacked IJJs [39, 40, 41, 42].

In this paper, we discuss the difference in inter-mesa coupling when two of
three mesas are operated either by CVB or CCB modes, based on the polarization
observation of EM waves emitted from mesas in individual and coupled emission
states. A CVB mostly generates a coupled emission with unimodal frequency
spectrum, whereas a CCB tends to generate an uncoupled emission with bimodal
frequency spectrum. The coupled emission state is a superposition of the
individual emissions perturbed by the inter-mesa coupling. The coefficients of
the linear combination with the bases of the individual oscillation states ,
which are chosen necessarily, compose the mesa interaction matrices, i.e., the
states of the harmonically oscillating plasmons inside the superconducting
substrate that are responsible for the interaction between the nonlinear
Josephson plasma oscillations in the mesas [43]. A systematic survey of the
interaction matrix will lead to the development of powerful THz sources by
controlling the mesa-to-mesa synchronization and will stimulate emergence of
superconducting devices that enable THz quantum communication.

##  II Experiment

In this study, we discuss EM wave radiations from three mesas (B, C, and E)
formed on a Bi2212 single crystal grown by the floating zone method using
photolithography and argon ion milling [19, 24]. The device is shown in Fig. 1
(a). The basic properties, such as the mesa shape, $T_{c}$, and
current–voltage characteristics of the three mesas, are summarized in the
Supplementary Materials. Parallel and series connections can simultaneously
energize a pair of mesa structures connected by a Bi2212 substrate single
crystal with CVB and CCB modes, respectively. For a CVB, the wires connected
to the two mesas were short-circuited at the outside of the cryostat, and a
voltage was applied between the electrode and the substrate crystal, so that
the same voltage was applied to the connected mesas, as shown in the right
panel of Fig. 1 (a). On the other hand for a CCB, a bias voltages was applied
between the two mesas, so it was not ensured that the same voltage was applied
to each mesa. An EM wave emitted was detected by a silicon bolometer cooled by
liquid helium [24], and the intensity was evaluated by the bolometer output
voltage and the radiation frequency measured using an FTIR spectrometer with a
split mirror[44, 15, 45]. For the polarization evaluation, the Stokes
polarization parameters (SPPs) were estimated from the transmitted intensity
through a polarization analyzer combining a fixed linear polarizer and a
rotating achromatic quarter wave plate (QWP) [46] as described in Refs. [47,
36, 48].

##  III Results and discussion

###  III.1 Current–voltage characteristics and emission intensity

Figure 1 (b) shows the current–voltage–emission ($I-V-E$) characteristics of
mesa B and E under CVB. This situation is represented by B$\parallel$E
hereinafter. In comparison to data for individual mesas B and E (Fig. 1(c) and
Fig. S1 in the Supplementary Materials), the current–voltage characteristics
(IVC) for B$\parallel$E is elongated twice with respect to the current, and
the emission intensity indicates more complicated multi-modal behavior as a
function of the total bias current (Fig. 1 (b) right panel). First, the $I-V$
curve for B$\parallel$E with increasing bias is explained. Below 25 mA, both B
and E are superconducting and their bias currents are equal. Between 25 and 50
mA, unequal currents flow in mesa B and E to keep CVB thus parts of IJJs of
the mesas become resistive. In the $E-I$ curve for B$\parallel$E, two
intensity maxima were observed: a broad maximum at $I=48$mA, and a rather
sharp maximum at $I=26$ mA. The former current is slightly higher than the sum
of the bias currents for the maximum emission intensity of both mesas B and E
($\simeq$ 22 mA, commonly). Here, the both mesas participate in the radiation
and an additional current $\simeq$ 2 mA is supplied. The latter current
approximately correspond to the sum of the bias current for the maximum
emission intensity of either mesa B or E and the current ($\simeq$ 3 mA) for
the same bias voltage region due to the multivalent IVC [15, 19]. One mesa
emits intensively with 22 mA, while the other does not under biasing at the
same voltage. As listed in Table 1, the maximum detected intensity $P_{max}$
for B$\parallel$E is slightly smaller than the sum of $P_{max}$’s for B and E
and the second maximum intensity for B$\parallel$E is very close to the
$P_{max}$ for B or E. Thus, we interpret that for B$\parallel$E both mesas B
and E contribute to the emission above 40 mA while only either one contributes
between 20 and 40 mA. The breakings of IV curve (at 40 and 18 mA) are
attributed to the switchings of the internal synchronous phase dynamics of
each mesas. This feature is consistent with a numerical calculation in a
similar system consist of two IJJ stacks connected in parallel [49]. In the
subsequent discussion, therefore, spectra and polarization data above 40 mA
for B$\parallel$E are employed for comparison between simultaneous and
individual emissions.

A similar $I-V-E$ plot for the CCB of B and E (B$-$E) is shown in Fig. 1(c)
together with a plot for B solo. The IVC for B$-$E below 1 V almost coincides
with the IVC for B. This is interpreted that only B contributes to the voltage
and a large part of mesa E remains zero voltage because of the considerable
difference in the distributions of the maximum Josephson currents of IJJs
composing the mesas B and E seen in Fig. S1. It was also found that the
bolometer responses for B and B$-$E between 15 and 35 mA with increasing
current are superposable. This clearly indicates that only B contributes to
the emission power. In the same current range, with decreasing current, on the
other hand, significant increase in bolometer response of B$-$E in comparison
to the individual B was observed. The maximum emission of B$-$E was attained
at $I=18.4$ mA and the second maximum was found at $I=26.4$ mA (indicated by
solid and open triangles, respectively). These currents are approximately 4 mA
lower and higher than the maximum radiating current of the individuals B and
E, respectively. The net mesa voltage for B$-$E at the maximum emission power
was $V=1.89$ V, which was slightly lower than the sum of the bias voltages for
the individual emissions. These shifts in bias current and voltage have been
evaluated theoretically [43]. It is also interesting to note that the
bolometer detection in decreasing current of B$-$E (and B) is not only large
but also highly fluctuating in comparison to that in increasing current. This
suggests that the coherent radiation is excited in the decreasing current
while partly incoherent or chaotic radiation [50, 51] may be excited in the
increasing current.

###  III.2 Spectral characteristics

The radiation spectra of B, C, E, B$\parallel$E, B$-$E, and B$-$C are shown in
Figs. 2(a) and (b). Unimodal spectra were observed not only for individual
operations but also for both CVB and CCB, suggesting that multiple mesas are
coupled and radiating synchronously. The observed unimodal spectra were as
follows: individual (B: 0.43–0.57, C: 0.50–0.55, E: 0.50–0.55 THz), B and E
(CVB: 0.47–0.53, CCB: 0.46–0.5 THz), and C and E (CVB: 0.47–0.53, CCB:
0.48–0.61 THz). The frequencies of the emission spectra as functions of the
averaged voltage applied to each mesa $V_{mesa}$ are shown in Fig. 2(c). The
linear fittings to the plots provide the number, $N$, of IJJs contributing to
the radiations. For the BE pair, in which $N$’s for individual emissions of B
and E are very close as $\sim 870$, and their effective $N$’s for CVB and CCB
radiations are approximately 810, which is 8 % smaller than those for B and E
individuals. Furthermore, radiation frequency region of B$\parallel$E and
B$-$E is close to the overlapping frequency range of B and E, which have
slightly shifted radiation frequency range due to the slight difference in
mesa width. Thus, the physics of EM radiation of the mesa B and E under
simultaneous bias can be reworded as synchronous oscillations of two
oscillators with slightly different eigenfrequencies with a cost of small
portion of oscillating Josephson junctions. For simultaneous bias of B and C
(CVB: 0.45–0.58, CCB: 0.43–0.55 THz), bimodal spectra were observed, as shown
in Fig. 2(b). At this time, the mesas are presumed to be radiating
independently. In particular, in the combination of B and C, a unimodal
spectrum is observed only at 0.45 THz for CVB, while bimodal spectra were
found in the other CVBs and all of the CCBs. The $N$ for C solo is distributed
between 780 and 810, which is more than 10 % smaller than that for B and E
individuals.

The result that CVB operations tend to be coupled is understood as follows.
For a CVB, an equal voltage is applied to the two mesas. Although it is not
promised to apply an equal voltage is to the stacked IJJs composing a mesa
structure, the inductive coupling among the IJJs[4] expanding the whole mesa-
stack results in an equal voltage being applied because of the Josephson
oscillation of an identical frequency. Consequently, an equal voltage is
applied to all contributing IJJs, and then Josephson plasma waves of a single
frequency are excited in the CVB mesas. The BE and CE pairs, for which a
unimodal spectrum is always observed, are presumed to have a relatively strong
coupling between the mesas.

It is intriguing that the coupling in the BE pair with larger separation is
relatively stronger than that in the combination of the BC pair resulting in
synchronous oscillations. The reasons for this are discussed first. The most
important factor is the commensurability of the inter-mesa separation $d$ with
respect to the mesa width $w$, which is considered as a dominant factor in the
emission frequency of a single mesa. When we consider that the coupling is
mediated by the Josephson plasmons, zero-voltage Josephson harmonic
oscillations inside the superconducting substrate underneath the mesas, which
has been observed as microwave absorption [52, 3], the interference between
the plasmons and the nonlinear Josephson oscillations excited in the mesas
determines the coupling. Here, using $w=67\mu$m (approximate average of widths
of the mesas listed in Table SI of the Supplementary Materials), $d/w$ values
of 1.3 (BC) and 6.1 (BE) are obtained. For combinations with $d/w$ closer to
an integer, the effective coupling attributed to the strength of the frequency
entrainment of the nonlinear Josephson plasma oscillation is more pronounced.
The second most important factor is the similarity in the device geometry,
including electrodes: mesas B and E are mirror-symmetric while C is neither
mirror- nor translational- symmetric with respect to B, and a large overlap in
the emission frequency regions of individually biased cases are observed. In
other words, for the cases that the device shapes are closer to being
equivalent and the emission frequencies are closer, even weak perturbations
may cause frequency entrainments, which result in synchronous radiations. In
summary, the BE pair provides coupled phase dynamics between two oscillators
with _close eigenfrequencies_.

###  III.3 Polarization properties

Next, we discuss polarization on radiations from mesas B, E, and their
combinations. Using the polarization analyzer described in the previous
section, we obtain the transmitted intensity as a function of the QWP angle
$\theta$ for B, E, B$-$E, and B$\parallel$E, as shown in Figs. 3(a) and (b).
Here, bias voltages to respective mesas are common as 0.92 V, for example.
Based on these results, the Stokes polarization parameters SPPs
$(S_{0},S_{1},S_{2},S_{3})=S_{0}(1,\tilde{S}_{1},\tilde{S}_{2},\tilde{S}_{3})$
can be estimated by fitting

$I\left(\theta\right)=\frac{1}{2}\left(S_{0}+S_{1}\cos^{2}{2\theta}+S_{2}\cos{2\theta}\sin{2\theta}+S_{3}\sin{2\theta}\right),$
(1)

to the experimental data. However, non-negligible two-fold asymmetry is found
in the experimental data despite of the two-fold rotationally symmetric
function of Eq. (1). This is attributed to an asymmetric loss of the QWP
possibly due to deformations of its metallic plates. See Sec. Siii in the
Supplementary Materials. In order to reduce this loss, we introduce a
correction function $C(\theta)$ as

$C(\theta)=1-\sqrt{a^{2}+b^{2}}+a\cos\theta+b\cos\theta,$ (2)

where $a$ and $b$ are constants determined by the optical system. We employ
$I_{corr}(\theta)=C(\theta)I(\theta)$ to fit the experimental data. This
operation corresponds to take a projection to original cross section of
intensity from observed results. The obtained SPPs for CVB and CCB are shown
in Figs. 3(c) and (d), respectively. The values are listed in Table SII in
Supplementary Materials. Between mesa B and E, no significant difference is
found for $S_{0},\tilde{S}_{1}$, and $\tilde{S}_{2}$, whereas $\tilde{S}_{3}$
has the opposite sign, i.e., opposite helicity. This is attributed to the
mirror symmetry of the device shapes including their electrodes. Next, let us
compare among individual and simultaneous bias cases. $S_{0}$’s for both CVB
and CCB cases considerably smaller than the square of the sum of
$\sqrt{S_{0}}$’s for individual cases. Significant difference in
$\tilde{S}_{3}$ is found between CVB and CCB. This is attributed to the phase
delay to synchronize B and E.

####  III.3.1 Linear combination method to describe a coupled emission

Let us consider the coupled emission of the BE pair. If we write electric
field emitted from the individually biased mesas B and E as
$\ket{{\text{B}}}=\bm{E}_{\text{B}}\exp[i(\omega_{\text{B}}t+\delta_{\text{B}})]$
and
$\ket{{\text{E}}}=\bm{E}_{\text{E}}\exp[i\left(\omega_{\text{E}}t+\delta_{\text{E}}\right)]$,
respectively, where $\bm{E}_{\text{B}}$, $\omega_{\text{B}}$, and
$\delta_{\text{B}}$ are the electric field vector, angular frequency, and the
phase of an EM wave emitted from mesa B, respectively. The simultaneous bias
in the parallel connection can be written as follows:

$\left|{\text{B}}\parallel{\text{E}}\right\rangle=\alpha\left|{\text{B}}\right\rangle+\beta\left|{\text{E}}\right\rangle,$
(3)

where $\alpha$ and $\beta$ are complex. This is based on the idea that the
observed coupled emission $\left|{\text{B}}\parallel{\text{E}}\right\rangle$
is a superposition of the EM waves emitted from mesas B and E under CVB,
referred as $\ket{\text{B}^{\prime}}$ and $\ket{\text{E}^{\prime}}$, which are
results of perturbation arising from the coupling interaction
$\mathrm{V}_{{\text{B}\parallel\text{E}}}$ to $\ket{\text{B}}$ and
$\ket{\text{E}}$, respectively. Thus,

$\binom{\ket{\text{B}^{\prime}}}{\ket{\text{E}^{\prime}}}=\mathrm{V}_{\text{B}\parallel\text{E}}\binom{\ket{\text{B}}}{\ket{\text{E}}}=\begin{pmatrix}\alpha_{1}&\beta_{1}\\\
\alpha_{2}&\beta_{2}\\\ \end{pmatrix}\binom{\ket{\text{B}}}{\ket{\text{E}}},$
(4)

where $\alpha=\alpha_{1}+\alpha_{2}$ and $\beta=\beta_{1}+\beta_{2}$ [36].
Writing
$\left|{\text{B}}\parallel{\text{E}}\right\rangle=\bm{E}_{{\text{B}}\parallel{\text{E}}}\exp[i\left(\omega_{{\text{B}}\parallel{\text{E}}}t+\delta_{{\text{B}}\parallel{\text{E}}}\right)]$,
we obtain
$\alpha\left(\omega_{\text{B}}\right)=|\alpha|\exp[i\left(\omega_{\text{B}}^{\prime}t+\delta_{\text{B}}^{\prime}\right)]$
and
$\beta\left(\omega_{\text{E}}\right)=|\beta|\exp[i\left(\omega_{\text{E}}^{\prime}t+\delta_{\text{E}}^{\prime}\right)]$
with
$\omega_{{\text{B}}\parallel{\text{E}}}=\omega_{\text{B}}+\omega_{\text{B}}^{\prime}=\omega_{\text{E}}+\omega_{\text{E}}^{\prime}$
and
$\delta_{{\text{B}}\parallel{\text{E}}}=\delta_{\text{B}}+\delta_{\text{B}}^{\prime}=\delta_{\text{E}}+\delta_{\text{E}}^{\prime}$.
Here, $\omega_{\text{B}}^{\prime}$ and $\delta_{\text{B}}^{\prime}$ are the
frequencies and phase changes of mesa B associated with synchronization, which
indicates the magnitude of the inter-mesa perturbation. It is considered that
$|\alpha|$ and $|\beta|$ correspond to the ratios of the number of IJJs
contributing to for the coupled emission ($N_{\text{B}}^{\prime}$ and
$N_{\text{E}}^{\prime}$) with respect to those for the individual emissions
($N_{\text{B}}$ and $N_{\text{E}}$); i.e.
$|\alpha|=N_{\text{B}}^{\prime}/N_{\text{B}}$ and etc.
$\delta_{\gamma}=\arg\left(\beta/\alpha\right)=\delta_{\text{E}}^{\prime}-\delta_{\text{B}}^{\prime}$
is the phase difference of the coupling perturbation for mesa B and E, which
may related to the separation between the two mesas and the wavelength of the
simultaneous oscillations of the two mesas. In the following, we derive

$({\alpha},{\beta})=\ket{\text{BE}}\binom{\ket{\text{B}}}{\ket{\text{E}}}^{-1},$
(5)

where $\ket{\text{BE}}$ is either $\ket{\text{B}\parallel{\text{E}}}$ or
$\ket{\text{B}-\text{E}}$, and the validity of determining $\alpha$ and
$\beta$ is discussed with fixing bases as specific polarizations of individual
emissions in the following section.

####  III.3.2 Fixed bias analysis results and discussion

First, let us discuss synchronization between two mesas in CVB mode. Since an
equal voltage is applied for both mesas in the CVB mode, two sets of SPPs of
$\ket{\text{B}}$, $\ket{\text{E}}$, and $\ket{\text{B}\parallel\text{E}}$ at
two constant voltages are employed and compare their coefficients $\alpha$ and
$\beta$ derived by Eq. (5). Here, we assume that the oscillation frequency of
the mesa does not change accompanied by the synchronization, i.e.,
$\omega_{\text{B}}^{\prime}=\omega_{\text{E}}^{\prime}=0$. The coefficients
for $V_{mesa}=$ 0.92 and 0.97 V are listed in Table 2. The set of
$(\alpha,\beta)$ at 0.92 V shows that the most of IJJs in both mesas B and E
contribute to the radiation with 20 degrees phase difference. Here, the
Josephson oscillation in the mesas B and E are apparently synchronized. On the
other hand at $V_{mesa}=$0.97 V, the obtained coefficients implies that only a
small portion of IJJs composing the mesa B participates in the radiation for
B$\parallel$E. Here, the phase difference $\delta_{\gamma}$ does not make
sense. This would be attributed larger deviation in radiation frequency of
B$\parallel$E from individual B. As an example of CCB mode, we discuss
$I_{mesa}=$ 30 mA, shown in Figs. 3 (b) and (d). The obtained $|\alpha|$ is
less than 0.3, suggesting that only 30 % of IJJs of the mesa B participates in
the synchronization. This seems to be consistent with the spectroscopic
analysis that the CCB mode tends to show a bimodal FTIR spectrum.

In our previous publications [36, 37], the bases used for the linear
combination analysis were regarded as the polarizations with the intensity
maximum of individual bias operations. A polarization of parallelly connected
mesas was composed from two polarizations of single mesas at biases determined
by radiation properties of respectively specified conditions. This procedure
is considered to be the same as previous works for synchronous radiations of
multiple mesas, where intensities and spectra at biases selected by their
authors with implicit aspects are compared to discuss correspondences between
individual and simultaneous bias operations[34, 35]. However, the requirement
of common voltage or current either for parallel or series connection has not
been necessarily taken into considerations.

Our analysis relies on three specific measured polarizations: a coupled
emission with respect to two individual emissions. Considering the known facts
on JPE, i.e., that emission intensity and polarization are sensitive to the
temperature distribution of the device [22], it is natural to presume that the
temperature distribution of the simultaneous emission is different from the
individual emissions. This discrepancy may result in deviations of $|\alpha|$
and $|\beta|$ from unity. Let us apply these bases to the neighboring
synchronous radiations to find frequency evolutions of the phase shift
$\delta_{\gamma}$. Here, the frequency shift due to the mesa interaction
$\omega^{\prime}$ is explicit. Figure 4 shows $\delta_{\gamma}$, $|\alpha|$,
and $|\beta|$ for CVB radiations with bases at 0.92 V used in the previous
paragraphs. Obtained parameters for CCB radiations with bases at 30 mA is also
plotted. It is found that the phase shift varies linearly with increasing
frequency as the general trend common for CVB and CCB. The expected phase
shift from the separation between the mesas $\delta_{d}=2\pi{ndf_{s}}/{c}_{0}$
is drawn with a solid line. Noted that $\delta$ obtained with using 0.97 V
bases shows no consistent trend. For CCB, the evolution also follows to
$\delta_{d}$ but the phase offset is different from CVB. The common slope in
the frequency evolution of $\delta_{\gamma}$ close to $\delta_{d}$ determined
by the refractive index of the superconducting substrate $n\simeq 4.1$
suggests that the interaction between the mesas are mediated not by
electromagnetic waves in the space but by plasmons inside the Bi2212 single
crystal. An expected phase delay in accordance of the pitch of the mesas
$\delta_{d+w}=2\pi{n(d+w)f_{s}}/{c}_{0}$ is also plotted in Fig. 4 (a). The
agreement to the experimental $\delta_{\gamma}$ looks less likely than the
case of $\delta_{d}$, however further analyses on synchronized radiations
among closer mesas (smaller $d/w$; C and D for the present device) are
required to judge the dominant distance factor ($d$ or $d+w$) for synchronized
oscillation between mesas.

The systematic difference between CVB and CCB observed in the present study
presents quantitative differences of media to promote the synchronous
radiation; i.e., the Josephson plasmon excited the base crystal. To attain
highest radiation power accompanied by fully controlled in-phase
synchronization, $\delta_{\gamma}$ holds an essential relevance for CVB, and
the number of synchronously oscillating IJJs within a mesa is another
relevance for CCB. In Ref. [38], it is reported that the phase delay between
two synchronized IJJ stacks composing the same number of IJJs is zero. The
difference from the present study may be attributed to small difference in
$N$, individual radiation frequency, inhomogeneous temperature rise, and so
on. These inevitable inequivalences of a real device can be compensated by an
externally applied perturbation such as an external magnetic field or a
focused laser irradiation. We believe that the suggested polarimetry under the
controlled perturbations and numerical analysis on Josephson junctions with
modified coupling capacitances [53] elucidate the mechanism for coherent phase
dynamics of a Josephson junction array. Furthermore, investigations on CVB and
CCB systems with more mesas in combination with artificially-fabricated
Josephson junction arrays [54] leads to multi-pixel two-dimensional JPE mesa
array devices and a global coupling system of Josephson oscillators [55] which
represents synchronous phenomena within a mesa consisting of thousands IJJs.

##  IV Summary

Terahertz electromagnetic wave emission from two simultaneously biased mesas
of Bi2212 IJJ stacks on a superconducting substrate was investigated by
intensity, frequency, and polarization measurements as functions of bias
voltage, combination of mesas, and parallel/series connections. Coupled and
uncoupled emissions tended to be observed in parallel and series connections
of the mesas, respectively. The frequencies of the coupled emissions were
systematically varied. Electromagnetic waves from pairs of coupled mesas can
be represented by linear combinations of two individual emissions at shifted
frequencies and phases, irrespective of parallel and series connections,
rather than by simple superpositions of observed individual radiations
modified by device temperature rise. Further systematic investigations with
parameters of inter-mesa separation and mesa geometries are required to fully
assess the features of the perturbation matrix to control the inter-mesa
synchronous phenomena and attain a high-power terahertz source from
superconducting IJJ stacks.

###### Acknowledgements.

This work was supported by the Japan Society for the Promotion of Science
(JSPS) KAKENHI (Grant No. 20H02606) and JSPS – Centre national de la recherche
scientifique (CNRS) Bilateral Program (Grant No. 120192908).

## References

  * Kleiner and Müller [1994] R. Kleiner and P. Müller, Physical Review B 49, 1327 (1994).
  * Tachiki  _et al._ [1994] M. Tachiki, T. Koyama, and S. Takahashi, Phys. Rev. B 50, 7065 (1994).
  * Kakeya  _et al._ [1998] I. Kakeya, K. Kindo, K. Kadowaki, S. Takahashi, and T. Mochiku, Physical Review B 57, 3108 (1998).
  * Sakai  _et al._ [1993] S. Sakai, P. Bodin, and N. F. Pedersen, J. Appl. Phys. 73, 2411 (1993).
  * Koyama and Tachiki [1996] T. Koyama and M. Tachiki, Phys. Rev. B 54, 16183 (1996).
  * Machida  _et al._ [1999] M. Machida, T. Koyama, and M. Tachiki, Physical Review Letters 83, 4618 (1999).
  * Machida and Sakai [2004] M. Machida and S. Sakai, Phys. Rev. B 70, 144520 (2004).
  * Machida  _et al._ [2000] M. Machida, T. Koyama, A. Tanaka, and M. Tachiki, Physica C: Superconductivity and its Applications 330, 85 (2000), arXiv:9907136 [cond-mat] .
  * Tachiki  _et al._ [2005] M. Tachiki, M. Iizuka, K. Minami, S. Tejima, and H. Nakamura, Physical Review B - Condensed Matter and Materials Physics 71, 134515 (2005), arXiv:0407052 [cond-mat] .
  * Bulaevskii and Koshelev [2007] L. N. Bulaevskii and A. E. Koshelev, Physical Review Letters 99, 057002 (2007).
  * Ozyuzer  _et al._ [2007] L. Ozyuzer, A. E. Koshelev, C. Kurter, N. Gopalsami, Q. Li, M. Tachiki, K. Kadowaki, T. Yamamoto, H. Minami, H. Yamaguchi, T. Tachiki, K. E. Gray, W.-K. Kwok, and U. Welp, Science (New York, N.Y.) 318, 1291 (2007).
  * Kadowaki  _et al._ [2008] K. Kadowaki, H. Yamaguchi, K. Kawamata, T. Yamamoto, H. Minami, I. Kakeya, U. Welp, L. Ozyuzer, A. Koshelev, C. Kurter, K. Gray, and W.-K. Kwok, Physica C: Superconductivity and its Applications 468, 634 (2008).
  * Minami  _et al._ [2009] H. Minami, I. Kakeya, H. Yamaguchi, T. Yamamoto, and K. Kadowaki, Applied Physics Letters 95, 232511 (2009).
  * Hu and Lin [2010] X. Hu and S.-Z. Lin, Superconductor Science and Technology 23, 053001 (2010), arXiv:0911.5371 .
  * Wang  _et al._ [2010] H. B. Wang, S. Guénon, B. Gross, J. Yuan, Z. Jiang, Y. Zhong, M. Grünzweig, A. Iishi, P. Wu, T. Hatano, D. Koelle, and R. Kleiner, Physical Review Letters 105, 057002 (2010).
  * Yurgens [2011] A. Yurgens, Physical Review B 83, 184501 (2011).
  * Kashiwagi  _et al._ [2011] T. Kashiwagi, K. Yamaki, M. Tsujimoto, K. Deguchi, N. Orita, T. Koike, R. Nakayama, H. Minami, T. Yamamoto, R. a. Klemm, M. Tachiki, and K. Kadowaki, Journal of the Physical Society of Japan 80, 094709 (2011).
  * Koyama  _et al._ [2011] T. Koyama, H. Matsumoto, M. Machida, and Y. Ota, Superconductor Science and Technology 24, 085007 (2011).
  * Kakeya  _et al._ [2012] I. Kakeya, Y. Omukai, T. Yamamoto, K. Kadowaki, and M. Suzuki, Applied Physics Letters 100, 242603 (2012).
  * Welp  _et al._ [2013] U. Welp, K. Kadowaki, and R. Kleiner, Nature Photonics 7, 702 (2013).
  * Li  _et al._ [2012] M. Li, J. Yuan, N. Kinev, J. Li, B. Gross, S. Guénon, A. Ishii, K. Hirata, T. Hatano, D. Koelle, R. Kleiner, V. P. Koshelets, H. Wang, and P. Wu, Physical Review B 86, 060505 (2012).
  * Tsujimoto  _et al._ [2014] M. Tsujimoto, H. Kambara, Y. Maeda, Y. Yoshioka, Y. Nakagawa, and I. Kakeya, Physical Review Applied 2, 044016 (2014).
  * Hao  _et al._ [2015] L. Y. Hao, M. Ji, J. Yuan, D. Y. An, M. Y. Li, X. J. Zhou, Y. Huang, H. C. Sun, Q. Zhu, F. Rudau, R. Wieland, N. Kinev, J. Li, W. W. Xu, B. B. Jin, J. Chen, T. Hatano, V. P. Koshelets, D. Koelle, R. Kleiner, H. B. Wang, and P. H. Wu, Physical Review Applied 3, 024006 (2015).
  * Kakeya and Wang [2016] I. Kakeya and H. Wang, Superconductor Science and Technology 29, 073001 (2016).
  * Kashiwagi  _et al._ [2017] T. Kashiwagi, H. Kubo, K. Sakamoto, T. Yuasa, Y. Tanabe, C. Watanabe, T. Tanaka, Y. Komori, R. Ota, G. Kuwano, K. Nakamura, T. Katsuragawa, M. Tsujimoto, T. Yamamoto, R. Yoshizaki, H. Minami, K. Kadowaki, and R. A. Klemm, Superconductor Science and Technology 30, 074008 (2017).
  * Borodianskyi and Krasnov [2017] E. A. Borodianskyi and V. M. Krasnov, Nature Communications 8, 1742 (2017).
  * Elarabi  _et al._ [2017] A. Elarabi, Y. Yoshioka, M. Tsujimoto, and I. Kakeya, Physical Review Applied 8, 064034 (2017).
  * Elarabi  _et al._ [2018] A. Elarabi, Y. Yoshioka, M. Tsujimoto, and I. Kakeya, Applied Physics Letters 113, 052601 (2018).
  * Kleiner and Wang [2019] R. Kleiner and H. Wang, Journal of Applied Physics 126, 171101 (2019).
  * Benseman  _et al._ [2019] T. M. Benseman, A. E. Koshelev, V. Vlasko-Vlasov, Y. Hao, U. Welp, W.-K. Kwok, B. Gross, M. Lange, D. Koelle, R. Kleiner, H. Minami, M. Tsujimoto, and K. Kadowaki, Physical Review B 100, 144503 (2019).
  * Kuwano  _et al._ [2020] G. Kuwano, M. Tsujimoto, Y. Kaneko, T. Imai, Y. Ono, S. Nakagawa, S. Kusunose, H. Minami, T. Kashiwagi, K. Kadowaki, Y. Simsek, U. Welp, and W.-K. Kwok, Physical Review Applied 13, 014035 (2020).
  * Ono  _et al._ [2020] Y. Ono, H. Minami, G. Kuwano, T. Kashiwagi, M. Tsujimoto, K. Kadowaki, and R. A. Klemm, Physical Review Applied 13, 064026 (2020).
  * Saito  _et al._ [2021] Y. Saito, S. Adachi, R. Matsumoto, M. Nagao, S. Fujita, K. Hayama, K. Terashima, H. Takeya, I. Kakeya, and Y. Takano, Applied Physics Express 14, 033003 (2021).
  * Orita  _et al._ [2010] N. Orita, H. Minami, T. Koike, T. Yamamoto, and K. Kadowaki, Physica C: Superconductivity and its Applications 470, S786 (2010).
  * Benseman  _et al._ [2013] T. M. Benseman, K. E. Gray, A. E. Koshelev, W.-K. Kwok, U. Welp, H. Minami, K. Kadowaki, and T. Yamamoto, Applied Physics Letters 103, 022602 (2013).
  * Tsujimoto  _et al._ [2020] M. Tsujimoto, S. Fujita, G. Kuwano, K. Maeda, A. Elarabi, J. Hawecker, J. Tignon, J. Mangeney, S. Dhillon, and I. Kakeya, Physical Review Applied 13, 051001 (2020), arXiv:2004.07422 .
  * Hayama  _et al._ [2020] K. Hayama, S. Fujita, Y. Kuriyama, K. Maeda, M. Tsujimoto, and I. Kakeya, in  _2020 International Conference on UK-China Emerging Technologies, UCET 2020_ (2020).
  * Lin [2014] S.-Z. Lin, Journal of Applied Physics 115, 173901 (2014), arXiv:arXiv:1401.2271v3 .
  * Ota  _et al._ [2009] K. Ota, K. Hamada, R. Takemura, M. Ohmaki, T. Machi, K. Tanabe, M. Suzuki, A. Maeda, and H. Kitano, Physical Review B 79, 134505 (2009).
  * Kitano  _et al._ [2016] H. Kitano, Y. Takahashi, D. Kakehi, H. Yamaguchi, S.-i. Koizumi, and S.-y. Ayukawa, Journal of the Physical Society of Japan 85, 054703 (2016).
  * Nomura  _et al._ [2015] Y. Nomura, T. Mizuno, H. Kambara, Y. Nakagawa, and I. Kakeya, Journal of the Physical Society of Japan 84, 013704 (2015).
  * Nomura  _et al._ [2019] Y. Nomura, R. Okamoto, T.-a. Mizuno, S. Adachi, T. Watanabe, M. Suzuki, and I. Kakeya, Physical Review B 100, 144515 (2019).
  * Lin and Koshelev [2013] S.-Z. Lin and A. E. Koshelev, Physica C: Superconductivity 491, 24 (2013).
  * Eisele  _et al._ [2007] H. Eisele, M. Naftaly, and J. R. Fletcher, Measurement Science and Technology 18, 2623 (2007).
  * Kakeya  _et al._ [2013] I. Kakeya, N. Hirayama, T. Nakagawa, Y. Omukai, and M. Suzuki, Physica C: Superconductivity 491, 11 (2013).
  * Nagai  _et al._ [2015] M. Nagai, N. Mukai, Y. Minowa, M. Ashida, T. Suzuki, J. Takayanagi, and H. Ohtake, Optics Express 23, 4641 (2015).
  * Schaefer  _et al._ [2007] B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, American Journal of Physics 75, 163 (2007).
  * Russell A. Chipman, Wai-Sze Tiffany Lam [2019] G. Y. Russell A. Chipman, Wai-Sze Tiffany Lam,  _Polarized light and optical systems_ (CRC Press, 2019) p. 251.
  * Gross  _et al._ [2015] B. Gross, F. Rudau, N. Kinev, M. Tsujimoto, J. Yuan, Y. Huang, M. Ji, X. J. Zhou, D. Y. An, a. Ishii, P. H. Wu, T. Hatano, D. Koelle, H. B. Wang, V. P. Koshelets, and R. Kleiner, Superconductor Science and Technology 28, 055004 (2015).
  * Gulevich  _et al._ [2019] D. R. Gulevich, V. P. Koshelets, and F. V. Kusmartsev, Physical Review B 99, 060501 (2019).
  * Shahverdiev [2021] E. M. Shahverdiev, Journal of Superconductivity and Novel Magnetism 34, 1125 (2021).
  * Matsuda  _et al._ [1995] Y. Matsuda, M. Gaifullin, K. Kumagai, K. Kadowaki, and T. Mochiku, Physical Review Letters 75, 4512 (1995).
  * Ottaviani  _et al._ [2009] I. Ottaviani, M. Cirillo, M. Lucci, V. Merlo, M. Salvato, M. G. Castellano, G. Torrioli, F. Mueller, and T. Weimann, Physical Review B 80, 174518 (2009).
  * Galin  _et al._ [2020] M. Galin, F. Rudau, E. Borodianskyi, V. Kurin, D. Koelle, R. Kleiner, V. Krasnov, and A. Klushin, Physical Review Applied 14, 024051 (2020).
  * Wiesenfeld  _et al._ [1998] K. Wiesenfeld, P. Colet, and S. Strogatz, Physical Review E 57, 1563 (1998).

Table 1:  Measured properties of coupled emissions for B$\parallel$E, B$-$E, C$\parallel$E, and C$-$E. $P_{max}$ is the maximum detected radiation power represented bolometer response, $I_{max}$ and $V_{max}$ are current and voltage yielding $P_{max}$, respectively. $N$ is number of IJJs contributing to the maximum radiation power averaged to a mesa for comparisons.  | B | C | E | B$\parallel$E | B$-$E
---|---|---|---|---|---
$P_{max}$ (mV) | 0.418 | 0.215 | 0.468 | 0.737 | 0.799
$I_{max}$ (mA) | 21.9 | 33.8 | 21.7 | 47.3 | 18.3
$V_{max}$ (V) | 0.983 | 1.01 | 0.963 | 0.903 | 1.899
$N$ | 867 | 796 | 879 | 798 | 815
Table 2:  Obtained linear-coupling coefficients $\alpha,\beta$, and $\delta_{\gamma}=\arg(\beta/\alpha)$.  |  | Bias | $|\alpha|$ | $|\beta|$ |  $\arg(\beta/\alpha)$ (deg.)
---|---|---|---|---|---
(a) | B$\parallel$E | 0.92 V | 0.632 | 0.824 | 17.4
(b) | B$\parallel$E | 0.97 V | 0.145 | 0.953 | 306.1
(c) | B$-$E | 30 mA | 0.297 | 1.23 | -9.02

Figure 1: (a) A microscope picture of the device (left) and a schematic
drawing of the device and their connections for B $\parallel$ E
(right,magenta) and B $-$ E (cyan). Properties of mesa D is not dealt in this
paper. Separations between mesas ($d$) are 87.5 (BC) and 412 (BE) $\mu$m.
(b,c) Current-voltage characteristics and bolometer response of B$\parallel$E
(b, color-coded), B (c, red), and B $-$ E (c, cyan). Solid and open triangles
in the right panel of (b) and (c) point the first maximum and the second
maximum in radiation intensity mentioned in the text, respectively.  Figure 2:
FTIR spectra of coupled simultaneous radiations (a) and uncoupled simultaneous
radiations (b) together with individual radiations at vicinity mesa voltages.
In (c), peak frequencies of emission spectra are plotted as functions of
averaged mesa voltage $V_{mesa}$: net voltage divided by 2 for series
connections. Thick straight lines mean the ac Josephson relation with
corresponding number of IJJs, $N$. Thin gray lines are drawn for $N$’s with a
step of 20. All data are taken at bath temperature of 30 K.  Figure 3: (a,b)
Polar plots of transmission intensity of B, E, B$\parallel$E (a), and B$-$E(b)
through the polarization analyzer. Data at 0.92 V and 30 mA are plotted in (a)
and (b), respectively. Symbols represent measured intensity and curves are
least square fitting results of $I_{corr}(\theta)$ given by Eqs. (1) and (2)
to derive SPPs. (c,d) Derived SPPs from (a) and (b) are shown in (c) and (d),
respectively. $S_{0}$, $\tilde{S}_{1}$, and $\tilde{S}_{2}$ show no
significant difference among B, E, B$\parallel$E andB$-$E. $\tilde{S}_{3}$
representing helicity of EM vector has opposite sign between B and E. This is
due to the mirror symmetry of the device shape between them.  Figure 4:
Estimated phase delay $\delta_{\gamma}$ (a), $|\alpha|$ (b), and $|\beta|$ (c)
between the synchronized mesas as a function of frequency for CVB (square) and
CCB (circle) operations. Symbol size represents scaled $S_{0}$ and colors
represent averaged bias current $I_{mesa}$ for the synchronized radiation.
Solid lines are phase difference corresponding to the mesa separation $d$ and
mesa pitch $d+w$ of traveling waves inside a media with a refractive index
$n=4.1$. Arrows in (a) point data without frequency shift between individual
and synchronized radiations, which are listed at rows (a) and (c) of Table 2.


# Total, Equitable and Neighborhood Sum Distinguishing Total Colorings of Some
Classes of Circulant Graphs

S. Prajnanaswaroopa  Department of Mathematics, Amrita School of Engineering-
Coimbatore
Amrita Vishwa Vidyapeetham, India. {s_prajnanaswaroopa, j_geetha, s_
<EMAIL_ADDRESS><EMAIL_ADDRESS>J. Geetha  Department of
Mathematics, Amrita School of Engineering-Coimbatore
Amrita Vishwa Vidyapeetham, India. {s_prajnanaswaroopa, j_geetha, s_
<EMAIL_ADDRESS><EMAIL_ADDRESS>K. Somasundaram  Department of
Mathematics, Amrita School of Engineering-Coimbatore
Amrita Vishwa Vidyapeetham, India. {s_prajnanaswaroopa, j_geetha, s_
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: In this paper, we have obtained the total chromatic as well as
equitable and neighborhood sum distinguishing total chromatic numbers of some
classes of the Circulant graphs.

MSC Subject Classification: 05C15, 05C25, 05B15
Keywords: Total coloring; Equitable total coloring; Neighborhood sum
distinguishing total coloring; Cayley Graphs; Circulant graphs; Power of
Cycles.

##  1 Introduction

Let $G=(V(G),E(G))$ be a graph with the sets of vertices $V(G)$ and edges
$E(G)$ respectively. All the graphs considered here are finite, simple
connected and undirected graphs. Total coloring of a simple graph is a
coloring of the vertices and the edges such that any adjacent vertices or
edges or edges and their incident vertices do not receive the same color. The
total chromatic number of a graph G, denoted by $\chi^{\prime\prime}(G)$, is
the minimum number of colors that suffice in a total coloring. It is clear
that $\chi^{\prime\prime}(G)\geq\Delta(G)+1$, where $\Delta(G)$ is the maximum
degree of G.

The Total Coloring Conjecture (TCC) (Behzad [1] and Vizing [8] independently
proposed) is a famous open problem in graph theory that states that any simple
graph $G$ with maximum degree $\Delta(G)$ can be totally colored with at most
$\Delta(G)+2$ colors. The progress on this conjecture had been low in the
initial years but several advancements are in progress [4]. The graphs that
can be totally colored in $\Delta(G)+1$ colors are said to be type I and the
graphs with the total chromatic number $\Delta(G)+2$ is said to be type II.

Equitable total coloring is a total coloring of a graph such that any two
color classes of the total coloring differ by at most one element. The minimum
colors required in such a coloring of the graph is called the equitable total
chromatic number, denoted by $\chi^{\prime\prime}_{=}(G)$. Fu [3] conjectured
that for every graph $G$, $G$ has an equitable total $k$-coloring for each
$k\geq\text{ max }\\{\chi^{\prime\prime}(G),\Delta(G)+2\\}$. Later Wang [9]
proposed a weaker conjecture that $\chi^{\prime\prime}_{=}(G)\leq\Delta(G)+2.$

A proper total $k$ coloring of $G$ is called the neighborhood sum
distinguishing total coloring if $\sum_{c}(u)\neq\sum_{c}(v)$ for each edge
$uv$, where $\sum_{c}(u)$ is the sum of the color of the vertex $u$ and the
colors of edges incident with $u$. The minimum number of colors required is
called the neighborhood sum distinguishing total chromatic number, denoted by
$\chi^{\prime\prime}_{\Sigma}(G)$. Pilsniak and Wozniak [6] conjectured that
$\Delta(G)+2\leq\chi^{\prime\prime}_{\Sigma}(G)\leq\Delta(G)+3$ for any graph
$G$.

In an assignment of colors to the edges of a graph, a rainbow matching is a
set of independent edges such that each edge in the independent set has a
different color. For example, we can color the edges of any even cycle
$C_{{2n}}$ such that it has a rainbow perfect matching (an $n-$ coloring of
the edges with each perfect matching receiving $n$ colors is a rainbow
coloring).

A Cayley graph, Cay$(G,S)$, for a group $H$ with symmetric generating set
$S\subset H$ is a simple graph with vertices as all the elements of the group;
and edges between each two elements of the form $g$ and $gs$, where $g\in
H,s\in S$. Note that a symmetric subset implies $s\in S\implies s^{-1}\in S$.
The set $S$ is also supposed to not have the identity element of the group. A
circulant graph is a Cayley graph on the group $H=\mathbb{Z}_{n}$, the cyclic
group on $n$ vertices. A power of cycle $C_{n}^{k}$ is a circulant graph on
$\mathbb{Z}_{n}$ with generating set $\\{1,2,\ldots,k,n-k,\ldots,n-2,n-1\\}$.

In other words, given a sequence of positive integers $1\leq
d_{1}<d_{2}<...<d_{l}\leq\lfloor\frac{n}{2}\rfloor$, the circulant graph
$G=C_{n}(d_{1},d_{2},...,d_{l})$ has vertex set $V=Z_{n}=\\{0,1,2,...,n-1\\}$,
two vertices $x$ and $y$ being adjacent iff $x=(y\pm d_{i})\mod n$ for some
$i,1\leq i\leq l$ and a graph is a power of cycle, denoted $C^{k}_{n}$, $n$
and $k$ are integers, $1\leq k<\lfloor\frac{n}{2}\rfloor$, if
$V(C^{k}_{n})=\\{v_{0},v_{1},...,v_{n-1}\\}$ and $E(C^{k}_{n})=E^{1}\cup
E^{2}\cup...\cup E^{k}$, where
$E^{i}=\\{e_{0}^{i},e_{1}^{i},...,e^{i}_{n-1}\\}$ and
$e_{j}^{i}=(v_{j},v_{(j+i)\mod\ n})$, $0\leq j\leq n-1$ and $1\leq i\leq k$.

Campos and de Mello [2] verified the TCC for power of cycle $C_{n}^{k},n$ even
and $2<k<\frac{n}{2}.$ Also, they showed that one can obtain a
$\Delta(C_{n}^{k})+2$-total coloring for these graphs in polynomial time. They
also proved that $C_{n}^{k}$ with $n\cong 0\mod(\Delta(C_{n}^{k})+1)$ are
type-I and they proposed the following conjecture.

######  Conjecture 1.1.

Let $G=C_{n}^{k}$, with $2\leq k<\lfloor\frac{n}{2}\rfloor$. Then,

$\chi^{\prime\prime}(G)=\begin{cases}\Delta(G)+2,&\text{ if
}k>\frac{n}{3}-1\text{ and }n\text{ is \ odd}\\\ \Delta(G)+1,&\text{
otherwise}.\end{cases}$

Geetha et al. [5] proved the Campos and de Mello’s conjecture for some classes
of powers of cycles. Also, they verify the TCC for complement of powers of
cycles.

A Latin square is an $n\times n$ array consisting of $n$ entries of numbers
(or symbols) with each row and column containing only one instance of each
element. This means the rows and columns are permutations of one single $n$
vector with distinct entries. A Latin square is said to be commutative if it
is symmetric. A Latin square containing numbers is said to be idempotent if
each diagonal element contains the number equal to its row (column) number. In
addition, if the rows of the Latin square are just cyclic permutations (one-
shift of the elements to the right) of the previous row, then the Latin square
is said to be circulant (anti-circulant, if the cyclic permutations are
actually left shifts), the matrix (corresponds to the Latin square) can be
generated from a single row vector. The Latin square

1 | k+2 | 2 | k+3 | … | 2k+1 | k+1
---|---|---|---|---|---|---
k+2 | 2 | k+3 | 3 | … | k+1 | 1
… | … | … | … | … | … | …
… | … | … | … | … | … | …
k+1 | 1 | k+2 | 2 | … | k | 2k+1

is anti-circulant, commutative and idempotent. The entries of the square are
as follows:

$L=(l_{ij})=\begin{cases}m,\ \text{ if }\ i+j=2m\\\ k+1+m,\ \text{ if
}i+j=2m+1.\end{cases}$

From the above, it can be easily seen that the Latin square corresponding to
the matrix $L$ is commutative, idempotent and also anti-circulant. The
following lemma is due to Stong [7] (Corollary 2.3.1).

######  Lemma 1.1.

If $G$ is abelian and has even order then Cay$(G,S)$ is 1-factorizable.

##  2 Results on Powers of Cycles

######  Theorem 2.1.

Let $G=C_{n}^{k}$ be a power of cycle graphs and $(k+i)|n$ for some $1\leq
i\leq k+1$. Then $G$ satisfies TCC. In particular $G$ is type I if $n$ is
even.

###### Proof.

Case 1. $n$ is even.

Let $n\equiv 0\bmod(k+i)$ with $1\leq i\leq k+1$ and let us take $k+i=2m+1$.
We also know that there exists a commutative idempotent Latin square of odd
order, $k+i$ in this case, which we call $C^{\prime}$. Now, we consider two
tableau of the form
Tableau $B^{\prime}$:

$k+i+1$ $k+i+2$ $k+i+1$ $k+i+3$ $k+i+2$ $k+i+1$ … … … $k+i+1$ $2k+1$ … … …
$k+i+1$

Tableau $A^{\prime}$:

$2k+1$ $2k$ $2k-1$ $\ldots$ $k+i+1$ $2k+1$ $2k$ $\ldots$ $k+i+2$ $2k+1$ $2k$
$\ldots$ $2k+1$ $2k$ $2k+1$

Now, arranging the two tableau and the idempotent and commutative Latin square
of order $k+1$ in the below fashion, would give us the color matrix desired,
with $2k+1$ colors. The portion $C$ is the portion of the Latin square
$C^{\prime}$ which fits in the color matrix.

$C$ |  $B$ |  |  |  |  $A$
---|---|---|---|---|---
$B^{T}$ | C |  $A^{T}$ |  |  |
|  $A$ | C |  $B$ |  |
|  |  $B^{T}$ | C |  $A^{T}$ |
|  |  |  $A$ | C |  $B$
$A^{T}$ |  |  |  |  $B^{T}$ | $C$

In case $i=1$, the entries of $C^{\prime}$ are written wholly, so that the
tableau $A^{\prime}$ and $B^{\prime}$ are equal to the tableau $A$ and $B$. In
case $i>1$, the tableau $A^{\prime}$ and $B^{\prime}$ could be modified to
accommodate the missed numbers in the color matrix, which are deleted from the
commutative idempotent Latin square $C^{\prime}$, that is, the portion of the
$k+i$ Latin square starting from the $(k+2)^{nd}$ position in the first row,
$(k+3)^{rd}$ position in the second row and so on, is cut and juxtaposed on
the tableau $A^{\prime}$ and $B^{\prime}$ to give us the tableau $A$ and $B$.
In particular, if $i=k+1$, the tableau $A^{\prime}$ and $B^{\prime}$ are
wholly replaced with the portions deleted from the Latin square $C^{\prime}$.
The portion deleted from the Latin square, in case $i>1$ would be $D^{\prime}$
and its transpose, where $D$ given by:

$e_{1,k+2}$ $e_{1,k+3}$ $\ldots$ $\ldots$ $e_{1,k+i}$ $e_{2,k+3}$ $e_{2,k+4}$
$\ldots$ $e_{2,k+i}$ $\ldots$ $\ldots$ $e_{3,k+i}$ $\ldots$ $e_{4,k+i}$
$e_{k+i,k+i}$ ,

where $e_{ij}$ denote the entries of the Latin square $C^{\prime}$. Note that
the portion $A^{\prime}$ and $B^{\prime}$ are actually only edge coloring of
the subgraph formed after removing the edges of
$C_{n}^{\lfloor\frac{k+i-1}{2}\rfloor}$. The circulant graph with generating
set
$\\{\lfloor\frac{k+i-1}{2}\rfloor+1,\lfloor\frac{k+i-1}{2}\rfloor+2,\ldots,k,n-\lfloor\frac{k+i-1}{2}\rfloor-1,n-\lfloor\frac{k+i-1}{2}\rfloor-2,\ldots,n-k\\}$
is connected and the generating set is same for its group. Hence by Lemma 1.1,
the circulant graph is 1-factorable. Therefore the circulant graph is $d$ edge
colorable, where $d$ is the degree of the subgraph thereby ensuring the
compatibility of the various tableau and hence the total coloring of the
graph.

Case 2. $n$ is odd.
We first give a $(k+i)$-partial total coloring of the graph that corresponds
to totally coloring the subgraph
$C_{n}^{\left\lfloor\frac{k+i}{2}\right\rfloor}$ by using the similar concept
of the above case. That is, we first totally color the graph induced by the
first $k+i$ vertices as in a commutative idempotent Latin square of order
$k+i$ and repeat the same pattern for the remaining vertices, taken $k+i$ at a
time (as in above case) and then extend it to the full total coloring by
coloring the remaining edges, which can be always done in $2k-(k+i)+2=k-i+2$
colors by using the Vizing’s theorem. Hence, the total colors used is $2k+2$,
which thereby verifies TCC for such graphs. ∎

######  Example 2.1.

For example, we see that the graph $C_{21}^{6}$ can be totally colored by
first partially giving the part of the graph $C_{21}^{3}$ a type I total
coloring and then extending it to the whole graph by giving the remaining
edges a class 2 edge coloring, which is guaranteed by Vizing’s theorem.The
first part of the color matrix can be seen in Table 1.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 0 1 5 2 6 3 7 4 1 5 2 6 3
7 4 1 2 2 6 3 7 4 1 5 3 6 3 7 4 1 5 2 4 7 4 1 5 2 6 3 5 1 5 2 6 3 7 4 6 2 6 3
7 4 1 5 7 3 7 4 1 5 2 6 8 4 1 5 2 6 3 7 9 5 2 6 3 7 4 1 10 6 3 7 4 1 5 2 11 7
4 1 5 2 6 3 12 1 5 2 6 3 7 4 13 2 6 3 7 4 1 5 14 3 7 4 1 5 2 6 15 4 1 5 2 6 3
7 16 5 2 6 3 7 4 1 17 6 3 7 4 1 5 2 18 3 7 4 1 5 2 6 19 7 4 1 5 2 6 3 20 4 1 5
2 6 3 7

Table 1: Partial total Coloring of $C_{21}^{6}$.

The color matrix is then extended by giving a suitable $7$ coloring of the
edges of the remaining edges( which are the edges of the Cayley graph on
$\mathbb{Z}_{21}$ with generating set as $\\{4,5,6,15,16,17\\}$), which is
guaranteed by Vizing’s theorem. Hence, the total colors required for this
graph is $7+7=14$.

There are several classes of graphs satisfy the condition of Theorem 2.1. For
example, if $n$ were highly composite, we have several classes of such graphs.

The canonical total coloring of a complete graph of order $n$ is coloring the
vertices and edges of the complete graph in a particular fashion which we
describe here. First, we observe that the complete graph can be written as the
circulant graph with generating set consisting of all elements of the cyclic
group of order $n$ except $0$, that is $\\{1,2,\ldots,n-1\\}$. Now, the total
color matrix is filled as follows:

(i). The vertices are colored $1-2-\ldots-n$, corresponding to filling the
diagonal in the pattern $1-2-\ldots-n$.

(ii). For the first row of the color matrix, the colors corresponding to the
generator $s_{i}$ is $\frac{s_{i}}{2}+1$ if $s_{i}$ is even and
$\left\lceil\frac{n}{2}\right\rceil+s_{i}$ if $s_{i}$ is odd.

(iii). The remaining rows of the total color matrix are filled by the
circulant pattern, that is rotating the first row to the left by one color
cyclically.

Note that the above canonical coloring implies that if we have color $x$ in
the first row of the total color matrix at position $s_{i}$, we have the color
$x-s_{i}\pmod{n}$ at position $n-s_{i}$ in the first row.

######  Theorem 2.2.

The equitable total chromatic number of a power of cycle graph, $G=C_{n}^{k}$
with $(2k+1)|n$, $n$ is even, is equal to its total chromatic number. Further,
$\chi^{\prime\prime}_{\Sigma}(G)\leq 2k+3=\Delta(G)+3$.

###### Proof.

Let $G=C_{n}^{k}$ and $n=2m(2k+1).$ From the case 1 of the previous Theorem,
the graph $C_{n}^{k}$ is type I and it is easy to see that the equitable total
chromatic of $C_{n}^{k}$ is equal to its total chromatic number (we remove the
vertices in each color classes, the remaining graphs have a perfect matching).

Secondly, we see that the edges of the hamiltonian cycle formed by the
generator $1$ in the totally colored graph have two rainbow perfect matchings.
We then give two new colors to the two rainbow perfect matchings, which will
then give us a different sum of the colors incident at each vertex for each
adjacent pair of vertices (as the row/column sum for any consecutive $2k+1$
rows/ columns will be different in the total color matrix). Hence the number
of colors used in such a coloring is $2k+1+2=2k+3$. However, it may be
possible to further reduce the number of total colors required by $1$, a
possibility which we cannot rule out. Hence, the neighbourhood sum
distinguishing total chromatic number, $\chi^{\prime\prime}_{\Sigma}(G)\leq
2k+3$. ∎

######  Example 2.2.

Let us consider the total coloring of the graph $C_{18}^{4}$. In this case ,
we have $2(4)+1=9|18$ and the graph is type I. We could write its total color
matrix as in Table 2.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 6 2 7 3 8 4 9 5 1 6 2 7 3 8 4
9 5 1 2 2 7 3 8 4 9 5 1 6 3 7 3 8 4 9 5 1 6 2 4 3 8 4 9 5 1 6 2 7 5 4 9 5 1 6
2 7 3 8 6 5 1 6 2 7 3 8 4 9 7 6 2 7 3 8 4 9 5 1 8 7 3 8 4 9 5 1 6 2 9 8 4 9 5
1 6 2 7 3 10 9 5 1 6 2 7 3 8 4 11 1 6 2 7 3 8 4 9 5 12 2 7 3 8 4 9 5 1 6 13 3
8 4 9 5 1 6 2 7 14 8 4 9 5 1 6 2 7 3 15 4 9 5 1 6 2 7 3 8 16 9 5 1 6 2 7 3 8 4
17 5 1 6 2 7 3 8 4 9

Table 2: Equitable total coloring of $C_{18}^{4}$.

We modify the two rainbow perfect matchings formed by the generating elements
$\\{1,17\\}$ with two new colors as explained in the theorem to give a
coloring such that the neighbourhood sum distinguishing total chromatic number
is $11$ from the total chromatic number $9$. Hence, the Table 2 could be
modified to Table 3.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 10 2 7 3 8 4 9 11 1 10 2 11 3
8 4 9 5 1 2 2 11 3 10 4 9 5 1 6 3 7 3 10 4 11 5 1 6 2 4 3 8 4 11 5 10 6 2 7 5
4 9 5 10 6 11 7 3 8 6 5 1 6 11 7 10 8 4 9 7 6 2 7 10 8 11 9 5 1 8 7 3 8 11 9
10 1 6 2 9 8 4 9 10 1 11 2 7 3 10 9 5 1 11 2 10 3 8 4 11 1 6 2 10 3 11 4 9 5
12 2 7 3 11 4 10 5 1 6 13 3 8 4 10 5 11 6 2 7 14 8 4 9 5 11 6 10 7 3 15 4 9 5
1 6 10 7 11 8 16 9 5 1 6 2 7 11 8 10 17 11 1 6 2 7 3 8 10 9

Table 3: Neighborhood sum distinguishing total coloring of $C_{18}^{4}$.

##  3 Results on Circulant Graphs

In this section, we obtain upper bound of total coloring for certain classes
of circulant graphs.

######  Theorem 3.1.

Let $G$ be a circulant graph with even order $n$, $\Delta(G)\geq\frac{n}{2}$,
and the generating set $S$ satisfying the following properties:
(i)The set does not have the involution element $\frac{n}{2}.$
(ii)The set has a subset $S_{1}$ of the form
$\\{1,2,3\ldots,\frac{n}{4},n-\frac{n}{4},\ldots n-1\\}$ (which is a
generating set of the power of cycle $C_{n}^{\frac{n}{4}}$).
(iii) The complement of the subset $S_{1}$ in $S$ generates the whole group.
Then $G$ satisfies the TCC.

###### Proof.

We bifurcate the total color matrix into two parts, first one consisting of
partial total coloring of the graph induced by the generating set
$\\{1,2,3,\ldots,\frac{n}{4},\ldots,n-1\\}$. The second part consisting of
edge coloring of the remaining edges. The bifurcation of the graph is done by
using the generating set $S_{1}$ and which is a power of cycle. The second
graph is induced by $\overline{S_{1}}$.

Here, the first part of the total color matrix requires only $\frac{n}{2}+2$
colors by using the result of Campos and de Mello [2]. From Lemma 1.1, the
graph induced by the remaining edges form a Cayley graph and it is
1-factoriziable. Therefore we can color the remaining edges with
$\Delta(G)-\frac{n}{2}$ colors. This gives us the full total coloring of the
original circulant graph using
$\Delta(G)-\frac{n}{2}+\frac{n}{2}+2=\Delta(G)+2$ colors thus satisfying TCC.
∎

######  Example 3.1.

Let us consider the total coloring of the cayley graph on $\mathbb{Z}_{20}$
with generating set $\\{1,2,3,4,5,7,8,12,13,15,16,17,18,19\\}$. Here, we have
degree of the graph equal to $14$ . The first part consists of the graph
corresponding to the generating subset $\\{1,2,3,4,5,15,16,17,18,19\\}$. We
can follow the process given in [2] to give a total $12$ coloring of the
subgraph induced by this subset. Since the complement of the subset is
$\\{7,8,12,13\\}$ generates the whole group, therefore, we could color the
remaining edges with $4$ extra colors. This implies that we could give a total
coloring of $G$ in $12+4=16$ colors which is exactly $\Delta(G)+2$ colors
here, or, $G$ satisfies TCC.

######  Theorem 3.2.

If a circulant graph $G$ of even order $n$ has the generating set
$S=\\{s_{i}\\}$ with $|S|=\frac{n}{2}-2$ such that none of
$s_{i}+s_{j}\neq\frac{n}{2},\ s_{i},s_{j}\in S$ and $\frac{n}{2}\notin S$,
then $\chi^{\prime\prime}(G)\leq\frac{n}{2}+1=\Delta(G)+3.$

###### Proof.

We begin by noticing that we can partition the vertices into $\frac{n}{2}$
independent sets as
$\\{0,\frac{n}{2}\\},\\{1,\frac{n}{2}+1\\},\ldots,\\{\frac{n}{2}-1,n-1\\}$ as
$\frac{n}{2}\notin S$. We observe that the induced subgraphs formed by the
vertices $\\{0,1,\ldots,\frac{n}{2}-1\\}$ and $\\{\frac{n}{2},\ldots,n-1\\}$
are isomorphic. Now, we totally color the induced subgraphs using the total
coloring of a complete graph of order $\frac{n}{2}$ (we give a similar total
coloring to the edges between two vertices of the sub graph as in the complete
graph). In order to color the edges joining the two subgraphs, we observe that
the edges missed from the complete graph while coloring each subgraph is then
later covered in the joining edges between the subgraphs. This is so possible
because if we have an edge between two vertices, say $g_{i}$ and $g_{j}$, we
will not have an edge between $g_{j}$ and
$g_{i}+\frac{n}{2}\pmod{\frac{n}{2}}$ or vice-versa; as this would imply that
both $g_{j}-g_{i}\pmod{\frac{n}{2}}$ and
$g_{i}+\frac{n}{2}-g_{j}=\frac{n}{2}-(g_{j}-g_{i})\pmod{\frac{n}{2}}$ belong
to the generating set, which is in contradiction to the assumption that we
have none of $s_{i}+s_{j}\neq\frac{n}{2}$. Therefore, we could give the color
to the edge between $g_{i}$ and $g_{j}$ which is missed from the complete
graph to the edge between $g_{i}$ and $g_{j}+\frac{n}{2}\pmod{\frac{n}{2}}$ in
$G$. Hence, we could color totally $G$ using the same colors as in the
complete graph. Since we require $\frac{n}{2}+1$ colors to totally color a
complete graph of order $\frac{n}{2}$, we are done. ∎

######  Example 3.2.

Consider the circulant graph on $\mathbb{Z}_{24}$ with generating set
$\\{1,3,4,5,10,14,19,\\\ 20,21,23\\}$. We see that it satisfies the conditions
of the theorem and hence can be totally colored with $13$ colors. The total
color matrix in this case can be written as in Table 4.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 1 8 9 3 10 6 2
11 5 12 7 1 8 2 9 10 4 11 7 3 12 6 13 2 9 3 10 11 5 12 2 4 13 7 1 3 9 10 4 11
12 6 13 3 5 1 8 4 3 10 11 5 12 13 7 1 4 6 2 5 10 4 11 12 6 13 1 8 2 5 7 6 11 5
12 13 7 1 2 9 3 6 8 7 12 6 13 1 8 2 3 10 11 7 9 8 13 7 1 2 9 3 4 5 12 8 10 9 1
8 2 3 10 4 12 6 13 9 11 10 6 2 9 3 4 11 5 13 7 1 10 11 7 3 10 4 5 12 13 1 8 2
11 12 2 11 5 12 1 8 9 3 10 6 13 3 12 6 13 2 9 10 4 11 7 14 2 4 13 7 1 3 10 11
5 12 15 3 5 1 8 9 4 11 12 6 13 16 4 6 2 3 10 5 12 13 7 1 17 5 7 10 4 11 6 13 1
8 2 18 6 8 11 5 12 7 1 2 9 3 19 11 7 9 12 6 13 8 2 3 10 20 5 12 8 10 13 7 1 9
3 4 21 12 6 13 9 11 1 8 2 10 4 22 13 7 1 10 6 2 9 3 11 5 23 7 1 8 2 11 7 3 10
4 12

Table 4: Total Coloring of the circulant graph in Example 3.2.

######  Theorem 3.3.

If a circulant graph $G$ of even order $n$ has a generating set $S$ such that
the following properties hold:
(i) It does not have the involution element $\frac{n}{2}.$
(ii) It has a subset $M$ of $S$ such that $|M|=\frac{n}{2}-2$ and
$m_{i}+m_{j}\neq\frac{n}{2},\ m_{i},m_{j}\in M.$
(iii) The complement of $M$ in $S$ generates the whole group.
Then, the graph $G$ satisfies $\chi^{\prime\prime}(G)\leq\Delta(G)+3$.

###### Proof.

From the previous theorem, we see that the total chromatic number of the
subgraph induced by the generating set $M$ is bounded above by
$\frac{n}{2}+1$. Now, as the complement of $M$ in $S$ generates the whole
group; From Lemma 1.1, the graph induced by the remaining edges form a Cayley
graph and it is 1-factoriziable. Therefore we can color the remaining edges
with $\Delta(G)-(\frac{n}{2}-2)=\Delta(G)-\frac{n}{2}+2$ colors. Thus, the
number of colors required for the total coloring of $G$ is at most
$\frac{n}{2}+1+\Delta(G)-\frac{n}{2}+2=\Delta(G)+3$ colors. ∎

######  Example 3.3.

Consider the circulant graph on $\mathbb{Z}_{24}$ with generating set

$\\{1,2,3,4,5,7,10,14,17,19,20,21,23\\}$.We see that this graph satisfies the
conditions of the theorem. Here we have $M=\\{1,3,4,5,10,14,19,20,21,23\\}$.
From theorem 2.2, we see that the graph induced by the set $M$ requires $13$
colors. The complement of $M$ in $S$, $\\{2,7,17,22\\}$ has cardinality of
$4$, whence the extra colors required are just $4$. Hence the graph can be
totally colored with $17$ colors.

######  Theorem 3.4.

Let $G$ be a circulant graph with $n=2m$ with $m$ odd and $\Delta(G)>m$. If
the generating sets $S$ and $S_{1}\subset S$, $|S_{1}|=m-1$, have the
following properties:
(i). $s_{i}\not\equiv s_{j}\pmod{\left\lceil\frac{n}{4}\right\rceil}$ where
$s_{i},s_{j}\in S_{1}$, $\frac{n}{2}\notin S_{1}$.
(ii). $S_{1}$ has a group generator in it.
(iii). Complement of $S_{1}$ in $S$ generates the whole group.
Then $\chi^{\prime\prime}_{=}(G)=\Delta(G)+1$ and
$\chi_{\Sigma}(G)\leq\Delta(G)+3$.

###### Proof.

We first totally color the subgraph induced by the generating subset $S_{1}$.
For this, we first color the vertices with the canonical colors
$1,2,3,\ldots\frac{n}{2}$ which implies that we have two vertices in each
color class and the diagonal of the total color matrix is filled in the
pattern $1-2-\ldots\frac{n}{2}-1-2\ldots,\frac{n}{2}$. This is possible as
$\frac{n}{2}\notin S_{1}$. Now, to fill the subdiagonals, we use the same
pattern as for the diagonals with the starting colors being different. The
starting colors (entries in the first row) is chosen as follows:

For $s_{i}<\frac{n}{2}$, the entries corresponding to $s_{i}$ in the first row
is chosen by using the index of $s_{i}$
$(s_{i}\mod{\left\lceil\frac{n}{4}\right\rceil})$ in the canonical total
coloring of a complete graph of order $\frac{n}{2}$. As for the subdiagonals
starting from $s_{i}>\frac{n}{2}$ , we use the colors used for inverses of
$s_{i}\mod{\left\lceil\frac{n}{4}\right\rceil}$ in the canonical total
coloring of the complete graph.

This ensures that there will be no clashes between the edge colors (as
$s_{i}\not\equiv s_{j}\mod{\left\lceil\frac{n}{4}\right\rceil}$). This also
ensures that there is no clashes between the vertex and edge colors, because,
if there are clashes between vertex and edge colors for $s_{i}>\frac{n}{2}$,
there will be clashes at the colors between the vertices and subdiagonals
starting at $s_{i}<\frac{n}{2}$, which is clearly not possible as
$s_{i}\not\equiv s_{j}\pmod{\left\lceil\frac{n}{4}\right\rceil}$. Thus, we can
give a type I total coloring for this subgraph using $m$ colors. Now, the
complement of $S_{1}$ in $S$ generates the whole group and by Lemma 1.1, the
remaining edges of $G$ are one factoriziable. Therefore $G$ is type I.

Since, in the above total coloring, we have equitable coloring of vertices and
the graph is type I, each total color class having vertices has a perfect
matching of the remaining graph (the graph with the two vertices removed). The
total color classes having only edges have just one element less than the
total color classes with vertices and as such, the graph has an equitable
total coloring using $\Delta(G)+1$ colors as desired. Thus,
$\chi^{\prime\prime}_{=}(G)=\chi^{\prime\prime}(G)=\Delta(G)+1.$

In order to prove that $\chi_{\Sigma}(G)\leq\Delta(G)+3$, we first give a
$\Delta(G)+1$ total coloring of $G$ as above and then take a Hamiltonian
cycle, (since a group generator is there in the generating set of the graph by
property ii) which has two rainbow perfect matchings. We give two extra colors
to its edges. This ensures that the sum of colors of neighbors is different.
Hence, $\chi_{\Sigma}(G)\leq(\Delta(G)+1)+2=\Delta(G)+3$. ∎

######  Example 3.4.

We take the graph $G$ the circulant graph on $\mathbb{Z}_{18}$ with generating
set $S=\\{1,2,4,6,7,8,10,11,12,14,16,17\\}$. Here, $S_{1}$ is
$\\{1,2,4,6,12,14,16,17\\}$. We first give a $9$-total coloring of the
subgraph induced by $S_{1}$. To give a total $9$-coloring, we use the total
color matrix of the canonical coloring of a complete graph of order $9$. This
gives us the starting colors for the subdiagonals starting at $1,2,4,6$ to be
$6,2,3,4$. correspondingly the subdiagonals starting at $12,14,16,17$ to be
$7,8,9,5$ .The part of graph induced by $\\{7,8,12,13\\}$, which is also a
circulant graph, can be given a $4$ coloring ( from Corollary 2.3.1 of [7]).
This gives us a $13$ total coloring of the graph which is also equitable. Now,
we take the rainbow Hamiltonian cycle induced by the set $\\{1,19\\}$ and give
it a $2$\- coloring using two extra colors (replacing the original coloring),
which is possible as the cycle induced is Hamiltonian (and hence even). This
gives us a neighborhood sum distinguishing total coloring using $15$ colors as
the colors replaced in each row and/or column happens to be having a distinct
sum owing to the rainbow nature of the prior coloring. The total color matrix
of the subgraph corresponding to $S_{1}$ is given in Table 5.
The final total color matrix can be obtained by using the Lemma 1.1 and $4$
extra colors to give us an equitable $13$\- coloring of the final graph.

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 1 | 6 | 2 |  | 3 |  | 4 |  |  |  |  |  | 7 |  | 8 |  | 9 | 5
1 | 6 | 2 | 7 | 3 |  | 4 |  | 5 |  |  |  |  |  | 8 |  | 9 |  | 1
2 | 2 | 7 | 3 | 8 | 4 |  | 5 |  | 6 |  |  |  |  |  | 9 |  | 1 |
3 |  | 3 | 8 | 4 | 9 | 5 |  | 6 |  | 7 |  |  |  |  |  | 1 |  | 2
4 | 3 |  | 4 | 9 | 5 | 1 | 6 |  | 7 |  | 8 |  |  |  |  |  | 2 |
5 |  | 4 |  | 5 | 1 | 6 | 2 | 7 |  | 8 |  | 9 |  |  |  |  |  | 3
6 | 4 |  | 5 |  | 6 | 2 | 7 | 3 | 8 |  | 9 |  | 1 |  |  |  |  |
7 |  | 5 |  | 6 |  | 7 | 3 | 8 | 4 | 9 |  | 1 |  | 2 |  |  |  |
8 |  |  | 6 |  | 7 |  | 8 | 4 | 9 | 5 | 1 |  | 2 |  | 3 |  |  |
9 |  |  |  | 7 |  | 8 |  | 9 | 5 | 1 | 6 | 2 |  | 3 |  | 4 |  |
10 |  |  |  |  | 8 |  | 9 |  | 1 | 6 | 2 | 7 | 3 |  | 4 |  | 5 |
11 |  |  |  |  |  | 9 |  | 1 |  | 2 | 7 | 3 | 8 | 4 |  | 5 |  | 6
12 | 7 |  |  |  |  |  | 1 |  | 2 |  | 3 | 8 | 4 | 9 | 5 |  | 6 |
13 |  | 8 |  |  |  |  |  | 2 |  | 3 |  | 4 | 9 | 5 | 1 | 6 |  | 7
14 | 8 |  | 9 |  |  |  |  |  | 3 |  | 4 |  | 5 | 1 | 6 | 2 | 7 |
15 |  | 9 |  | 1 |  |  |  |  |  | 4 |  | 5 |  | 6 | 2 | 7 | 3 | 8
16 | 9 |  | 1 |  | 2 |  |  |  |  |  | 5 |  | 6 |  | 7 | 3 | 8 | 4
17 | 5 | 1 |  | 2 |  | 3 |  |  |  |  |  | 6 |  | 7 |  | 8 | 4 | 9
Table 5: Partial equitable total Coloring of the Circulant graph in Example
3.4.

To give a neighborhood sum distinguishing total coloring using $15$ colors, we
modify the prior total color matrix by replacing the colors of the Rainbow
Hamiltonian cycle induced by the set $\\{1,19\\}$ as in Table 6.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 0 1 14 2 3 4 6 8 9 15 1 14 2 15 3
4 5 7 9 1 2 2 15 3 14 4 5 6 8 1 3 3 14 4 15 5 6 7 9 2 4 3 4 15 5 14 6 7 8 1 5
4 5 14 6 15 7 8 9 2 6 4 5 6 15 7 14 8 9 1 3 7 5 6 7 14 8 15 9 1 2 1 4 8 6 7 8
15 9 14 1 2 3 9 7 8 9 14 1 15 2 3 4 10 6 8 9 1 15 2 14 3 4 5 11 7 9 1 2 14 3
15 4 5 6 8 1 2 3 15 4 14 5 6 9 2 3 4 14 5 15 6 7 14 8 1 3 4 5 15 6 14 7 15 9 2
4 5 6 14 7 15 8 16 9 1 3 5 6 7 15 8 14 17 15 1 2 4 6 7 8 14 9

Table 6: Partial total coloring of the circulant graph in Example 3.4 that is
neighborhood sum distinguishing.

The final total color matrix can be obtained by using Lemma 1.1 and $4$ extra
colors to give us a total coloring with $\chi^{\prime\prime}_{\Sigma}(G)=15$.

Acknowledgements:
The first author was supported by CSIR Fellowship (File No.
09/942(0020)/2020-EMR-I) .
The second and third authors were supported by SERB, India (No.SERB:
EMR/2017/001869).

## References

  * [1] Behzad, Mehdi. Graphs and their chromatic numbers. Michigan State University, 1965.
  * [2] Campos, C. N., and de Mello, Célia Picinin. “A result on the total colouring of powers of cycles.” Discrete Applied Mathematics 155.5 (2007): 585-597.
  * [3] Fu, H-L. “Some results on equalized total coloring.” Congressus Numerantium (1994): 111-120.
  * [4] Geetha,J., Narayanan, N. and Somasundaram, K. “Total Colourings-A survey.” arXiv preprint arXiv:1812.05833 (2018).
  * [5] Geetha, J., K. Somasundaram, and Hung-Lin Fu. ”Total colorings of circulant graphs.” Discrete Mathematics, Algorithms and Applications (2020): 2150050.
  * [6] Pilśniak, Monika, and Woźniak, Mariusz. “On the total-neighbor-distinguishing index by sums.” Graphs and Combinatorics 31.3 (2015): 771-782.
  * [7] Stong, Richard A. “On 1-factorizability of Cayley graphs.” Journal of Combinatorial Theory, Series B 39.3 (1985): 298-307.
  * [8] Vizing, Vadim G. “Some unsolved problems in graph theory.” Russian Mathematical Surveys 23.6 (1968): 125.
  * [9] Wang, Wei-Fan. “Equitable total coloring of graphs with maximum degree 3.” Graphs and Combinatorics 18.3 (2002): 677-685.