# A Full Accounting of the Visible Mass in SDSS MaNGA Disk Galaxies

Nitya Ravi Department of Physics & Astronomy, University of Rochester, 500
Joseph C. Wilson Blvd., Rochester, NY 14627  Kelly A. Douglass Department of
Physics & Astronomy, University of Rochester, 500 Joseph C. Wilson Blvd.,
Rochester, NY 14627  Regina Demina Department of Physics & Astronomy,
University of Rochester, 500 Joseph C. Wilson Blvd., Rochester, NY 14627

###### Abstract

We present a study of the ratio of visible mass to total mass in spiral
galaxies to better understand the relative amount of dark matter present in
galaxies of different masses and evolutionary stages. Using the velocities of
the H$\alpha$ emission line measured in spectroscopic observations from the
SDSS MaNGA DR17, we evaluate the rotational velocity of 5522 disk galaxies at
their 90% elliptical Petrosian radii, $R_{90}$. We compare this to the
velocity expected from the total visible mass, which we compute from the
stellar, H I, H2, heavy metal, and dust masses. H2 mass measurements are
available for only a small subset of galaxies observed in SDSS MaNGA DR17, so
we derive a parameterization of the H2 mass as a function of absolute
magnitude in the $r$-band using galaxies observed as part of SDSS DR7. With
these parameterizations, we calculate the fraction of visible mass within
$R_{90}$ that corresponds to the observed velocity. Based on statistically
analyzing the likelihood of this fraction, we conclude that the null
hypothesis (no dark matter) cannot be excluded at a confidence level better
than 95% within the visible extent of the disk galaxies. We also find that, by
including all of these mass components, star-forming disk galaxies contain
statistically the same ratio of visible-to-total mass, independent of
magnitude.

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##  1 Introduction

Current cosmological models indicate that the dominant component of matter in
the universe is dark matter (Planck Collaboration et al., 2020), which
manifests itself primarily through gravity. Dark matter is expected to have
minimal to no interaction with the electromagnetic force, therefore emitting
little to no light. It is also unlikely to participate in the strong
interaction, since otherwise it would be embedded in nuclei. It is currently
unclear whether or not dark matter engages in the weak interactions (Porter et
al., 2011, and references therin).

Phenomena such as gravitational lensing around galaxy clusters (Bartelmann,
2010, and references therin) and galaxy kinematics (e.g., Freeman, 1970;
Bosma, 1978; Carignan & Freeman, 1985; Salucci, 2019) contribute to the
observational evidence for dark matter across most scales in the universe.
Constraints from Big Bang nucleosynthesis (Fields & Sarkar, 2006) and detailed
measurements of the imprint of the Baryon Acoustic Oscillations on the
anisotropy of the Cosmic Microwave Background (Komatsu et al., 2011) strongly
suggest that dark matter is of a non-baryonic nature. Simulations based on
cold dark matter models are able to reproduce the current distribution of
galaxies (e.g., Springel et al., 2005), indicating that dark matter is likely
to be composed of heavy, weakly interacting particles. However, ground-based
experiments have failed to observe any effects associated with the passage of
such particles through normal matter (Boveia & Doglioni, 2018). Moreover,
results from the Large Hadron Collider exclude most models that offer
plausible candidates for dark matter (for the latest results see ATLAS
collaboration, 2023; Aad et al., 2023; CMS Collaboration, 2023; Tumasyan et
al., 2022). Hence, solving the puzzle of dark matter is one of the leading
problems currently faced by the scientific community.

Modern large-scale galaxy surveys offer high quality data that allow us to
reevaluate the astronomical evidence for the existence of dark matter. One of
the original sources of such evidence was galactic rotation curves (Rubin &
Ford, 1970; Rubin et al., 1980, 1985, 1982). These studies were based on
samples with low statistics, containing only about 20 galaxies. The expected
rotational velocities of galaxies were estimated based only on stellar mass
and did not include gas or dust. Since the 1980s, rotation curve analysis has
been performed on larger galaxy samples to study various galaxy properties.
Mathewson et al. (1992) analyzed long slit spectroscopy, where velocities were
measured along the semi-major axis of galaxies, to construct the rotation
curves of over 900 galaxies. Persic et al. (1996) analyzed the rotation curves
of the same sample and found that the stellar disk did not contain sufficient
matter to produce the observed rotation curve.

Other studies that construct rotation curves from long slit spectroscopy
(e.g., Catinella et al., 2006; Di Teodoro et al., 2021) support the
observation that rotation curves ubiquitously flatten at the outer radii of
galaxies and find that the stellar mass scales with the inferred mass of the
dark halos. More recently, studies have fit rotation curves to stellar and gas
velocity fields using integral field spectroscopy (e.g., de Blok et al., 2008;
Torres-Flores et al., 2011; Kalinova et al., 2017; Schmidt et al., 2023) for
tens to hundreds of galaxies to estimate the galaxies’ dynamical masses and
model dark matter halo profiles. Douglass et al. (2019) and Yoon et al. (2021)
each study the rotation curves of almost 2000 SDSS MaNGA galaxies using either
gas or stellar kinematics. Because of the large variations in galaxy
properties throughout these samples, one of the biggest short-comings of these
prior studies has been their limited statistical power.

We perform a rotation curve analysis for over 5500 galaxies in SDSS MaNGA Data
Release 17 (DR17; Abdurro’uf et al., 2022) to analyze the dark matter content
of spiral galaxies. In this paper, we re-evaluate the amount of dark matter
needed to explain the observed rotational velocities and revisit the
statistical significance of the null, i.e. “no dark matter,” hypothesis using
the high statistics afforded by SDSS MaNGA (Bundy et al., 2015).

We construct models of rotation curves for each galaxy using the H$\alpha$
emission line velocities measured across a galaxy’s surface. Based on the
rotational velocity we infer the value of the total (gravitational) mass and
compare it to the visible mass. A similar analysis was conducted on 1988
galaxies in SDSS MaNGA DR15 (Aguado et al., 2019) in Douglass & Demina (2022),
where visible mass was defined as the sum of the stellar and atomic hydrogen
masses. The ratios of visible to total mass for these galaxies were studied by
splitting the sample into three subsamples based on color-magnitude
classification and analyzing the mass ratios’ dependence on the luminosity,
gas-phase metallicity, and color-magnitude classification.

We improve on the earlier studies by defining the visible mass as the sum of
the stellar, neutral atomic hydrogen, molecular hydrogen, helium, heavy
metals, and dust masses. We present the ratio of visible to total mass as a
function of galaxy luminosity. For each galaxy in our sample, we construct a
statistical model that accounts for the statistical and systematic
uncertainties on the measured rotational velocity, as well as the
uncertainties on each of the visible mass components. Using this statistical
model, we evaluate the level of consistency of the observed rotational
velocities with the null, i.e. “no dark matter”, hypothesis.

The paper is structured as follows. In Section 2, we discuss the data
selection process. In Section 3, we describe the modeling of the rotation
curves and stellar mass distributions. In Section 4, we detail the estimation
of the mass components. In Section 5, we describe the statistical model. We
present the results in Section 6; the data and the results of the statistical
analysis are summarized in Table 3. We conclude in Section 7.

##  2 SDSS MaNGA DR17 and Galaxy Selection

We use the H$\alpha$ emission line velocity maps from SDSS MaNGA DR17
(Abdurro’uf et al., 2022) to model the rotation curves of spiral galaxies. The
SDSS MaNGA survey used integral field spectroscopy to measure spectra at
different points throughout a galaxy by placing an integral field unit (IFU)
on each galaxy. The IFU is a bundle of spectroscopic fibers arranged in a
hexagonal shape containing between 19 and 127 fibers and covers 12.5” to 32.5”
in diameter (Law et al., 2015). The light received by the fibers was fed to
two spectrographs with wavelength ranges 3600–6000Å and 6000–10300Å,
respectively, with a resolution of $\lambda/\Delta\lambda\sim 2000$ (Drory et
al., 2015).

SDSS MaNGA DR17 is the final data release for the MaNGA survey and contains
more than 10,000 nearby galaxies in the northern sky. The target selection
process prioritized maintaining a flat distribution in luminosity (Wake et
al., 2017), so the survey consists of three subsamples: the primary sample,
where the IFU covers out to $1.5R_{e}$, the secondary sample, where the IFU
covers out to $2.5R_{e}$, and the color-enhanced sample, which supplements the
primary sample with high-mass blue galaxies and low-mass red galaxies. In
order to check for possible systematic bias, we present the results of our
analysis for the entire data set and each of these individual subsamples,
referred to as MaNGA sample 1, 2, and 3, respectively.

We extract each galaxy’s rotation curve using the H$\alpha$ velocity map and
$g$-band-weighted mean flux map as processed by the MaNGA Data Analysis
Pipeline (DAP; Westfall et al., 2019). The stellar mass rotation curve is
extracted from the stellar mass density maps processed by Pipe3D (Sánchez et
al., 2016, 2018). Absolute magnitudes are obtained from version 1.0.1 of the
NASA-Sloan Atlas (NSA; Blanton et al., 2011). Distances are in units of
Mpc/$h$, where $h$ is the reduced Hubble constant defined by $H_{0}=100h$
km/s/Mpc.

####  2.0.1 SDSS DR7

SDSS Data Release 7 (DR7; Abazajian et al., 2009) observed approximately one
quarter of the northern sky in both photometry and spectroscopy. A dedicated
2.5-m telescope at the Apache Point Observatory in New Mexico with a wide-
field imager and a pair of double fiber-fed spectrometers was used to conduct
the multiband imaging and spectroscopic survey. Photometric data was taken in
the five SDSS filters: $u$, $g$, $r$, $i$, and $z$ (Fukugita et al., 1996;
Gunn et al., 1998). Using 320 fibers placed into fiber plug plates with a
minimum fiber separation of 55”, follow-up spectroscopic analysis was
performed on galaxy targets with Petrosian $r$-band magnitudes $m_{r}\leq
17.77$ and $r$-band Petrosian half light radii $\mu_{50}\leq 24.5$ mag
arcsec-2 (Lupton et al., 2001; Strauss et al., 2002). For SDSS DR7, the
spectrometers covered a wavelength range of 3800–9200Å with a resolution of
$\lambda/\Delta\lambda\sim 1800$ (Smee et al., 2013).

We make use of the photometric data (colors, color gradients, and inverse
concentration indices) for MaNGA galaxies available from the KIAS-VAGC
(Blanton et al., 2005; Choi et al., 2010). We use the global emission line
fluxes from the Portsmouth group galaxy properties catalog (Thomas et al.,
2013) to calculate the gas-phase metallicity.

####  2.0.2 H I observations

H I mass estimates are obtained from the H I–Manga Data Release 3 (Stark et
al., 2021). H I–MaNGA is a follow-up survey of MaNGA galaxies conducted on the
Robert C. Byrd Green Bank Telescope (GBT) in Green Bank, West Virginia. The
third data release of H I–MaNGA also includes a cross-match between the
Arecibo Legacy Fast ALFA (ALFALFA) survey performed at the Arecibo Observatory
in Arecibo, Puerto Rico, and MaNGA DR17 targets.

####  2.0.3 CO observations

H2 masses are inferred from measurements of the CO(1-0) line emission from two
surveys: the MaNGA ARO Survey of Targets (MASCOT) first data release
(Wylezalek et al., 2022) and the xCOLD GASS survey (Saintonge et al., 2017).
The MASCOT survey performs observations of MaNGA galaxies at the Arizona Radio
Observatory. The xCOLD GASS survey conducted CO(1-0) observations of SDSS
galaxies on the IRAM 30-meter telescope in Spain.

###  2.1 Color-magnitude classification

Figure 1: $\Delta(g-i)$ color gradient versus $u-r$ color for our sample of
SDSS MaNGA galaxies with stellar mass estimates, marked by their color-
magnitude diagram classification: open red circles for the red sequence, green
stars for the green valley, and blue crosses for the blue cloud. The black
boundary is the separation between early- and late-type galaxies as defined by
Choi et al. (2010).

As shown in Douglass & Demina (2022), a galaxy’s ratio of total to stellar
mass depends on the galaxy’s evolutionary stage. We therefore separate the
galaxies into three populations — blue cloud, green valley, and red sequence —
in the color-magnitude diagram (CMD) to better understand these relationships.
Galaxies in the blue cloud are typically fainter and more blue, while galaxies
in the red sequence are brighter and more red. It is believed that galaxies
transitioning between the blue cloud and red sequence occupy the green valley
(Martin et al., 2007).

We use the same method to classify the galaxies into one of these three
populations as used in Douglass & Demina (2022), where the classification is
based on the inverse concentration index, $c_{\text{inv}}$, color, $u-r$, and
color gradient, $\Delta(g-i)$. As shown in Fig. 1, galaxies that are part of
the red sequence are those that generally fall above and to the right of the
depicted boundary originally defined by Park & Choi (2005) (normal early-type
galaxies), while galaxies that are part of the blue cloud are those that
generally fall below and to the left of the boundary (late-type galaxies).
Galaxies that are part of the green valley are either those above the boundary
but with $u-r<2$ (blue early-type galaxies) or a high $c_{\text{inv}}$, or
those below the boundary with $\theta<20^{\circ}$, where

$\theta=\tan^{-1}\left(\frac{-\Delta(g-i)+0.3}{(u-r)-1}\right).$ (1)

See Douglass & Demina (2022) for a more detailed description of the CMD
classification.

In this study, we require our objects to be dominated by rotational motion
(described in Section 3 below). As a result, we expect few galaxies in our
sample to be in the red sequence. After visual inspection, we find that the
red sequence galaxies that are in our final sample appear to be either red
disk galaxies with little to no star formation (likely lenticulars) or
elliptical galaxies that are still supported by rotation.

##  3 Modeling of the rotation curves and stellar mass distribution

###  3.1 Modeling the velocity map

Figure 2: IFU (magenta hexagon) overlaid on RGB composite image of MaNGA
galaxy 8997–9102 (made with the SDSS Marvin python package by Cherinka et al.,
2019). The IFU does not cover the entire visible extent of the galaxy, as is
common for MaNGA observations.

Figure 3: Masks for the different velocity map models for example galaxy
10001–12701. The masks for models 1 and 2 is shown on the top left, the mask
for model 3 is shown in the center, and the mask for model 4 is shown on the
top right. The histogram on the bottom left shows the distribution of unmasked
spaxel velocities used in models 1 and 2. Model 3 masks spaxels outside of the
vertical dashed lines. The histogram on the bottom right shows the
distribution of unmasked spaxel velocity dispersions used in models 1 and 2.
Model 4 masks spaxels to the right of the vertical dashed line. Note that
masking the outlying spaxels in the velocity distribution shifts the velocity
gradient (indicated by the colormap) to that expected for a rotating disk
galaxy.

Figure 4: Example H$\alpha$ velocity map from the MaNGA DAP (first column),
our best fit model to the velocity map (second column), the residual between
the velocity map and our best-fit model (third column), and the deprojected
rotation curve for for the galaxy (fourth column).

We estimate a galaxy’s total dynamical mass using its H$\alpha$ velocity map
obtained from the SDSS MaNGA DAP. Only spaxels with a data quality bit of 0
and S/N $\geq 5$ are included in the analysis.

We also require that all galaxies have a smooth gradient with a maximum
“smoothness score” of 2.0 as described in Douglass et al. (2019). We restrict
the analysis to galaxies with a T-Type $>0$ (late-type galaxies) as classified
by the MaNGA Morphology Deep Learning DR17 Value Added Catalog (Domínguez
Sánchez et al., 2022).

Similar to both Douglass et al. (2019) and Douglass & Demina (2022), the
velocity map of each galaxy is fit to the rotation curve parameterization
defined in Barrera-Ballesteros et al. (2018),

$V(r)=\frac{V_{\text{max}}r}{(R^{\alpha}_{\text{turn}}+r^{\alpha})^{1/\alpha}}$
(2)

where $V(r)$ is the rotational velocity at a distance $r$ from the center of
the galaxy. The free parameters are $V_{\text{max}}$, the magnitude of the
velocity at which the rotation curve plateaus, $R_{\text{turn}}$, the radius
at which the rotation curve changes from increasing to flat, and $\alpha$,
which describes the sharpness of the curve. The extent of the MaNGA H$\alpha$
velocity maps and the radius to which we can measure rotational velocities is
limited by the visible extent of the galaxy. Rotation curves are only fit out
to the maximum radius, $R_{\text{max}}$, covered by the integral field unit
(IFU), the extent of which is shown for an example galaxy in Fig. 2.

Each galaxy’s systemic velocity, kinematic center, inclination angle, and
position angle are also free parameters in this fit, resulting in a total of
eight free parameters. When determining the best-fit model for each galaxy, we
make use of
$\chi^{2}=\Sigma((\text{data}-\text{model})/\text{uncertainty})^{2}$ and
$\chi^{2}_{\nu}$, where we normalize $\chi^{2}$ by the difference between the
number of unmasked spaxels in the velocity map and the number of free
parameters in the fit. We define four best-fit models as follows:

Model 1

The model with the smallest $\chi^{2}$.

Model 2

The model with the smallest residual, $\Sigma(\text{data}-\text{model})^{2}$.

Model 3

To help remove foreground artifacts from the analysis, we define upper and
lower velocity bounds by binning all unmasked spaxels with a bin width of 10
km/s. The velocity bounds are defined as the nearest empty bin on either side
of the bin with the most spaxels, as shown in the histogram on the bottom left
of Fig. 3. Spaxels with values outside of this velocity range are masked; see
the top center of Fig. 3 for an example of the resulting mask. We then select
the model with the smallest $\chi^{2}$.

Model 4

To help remove spaxels that are potentially contaminated by emission from AGN,
which are defined as bins with an unusually high velocity dispersion, we
define an upper limit on the velocity dispersion by binning the velocity
dispersion of the unmasked spaxels with a bin width of 10 km/s. The velocity
dispersion upper bound is defined as the nearest empty bin to the lowest
velocity dispersion bin containing spaxels, as shown in the histogram on the
bottom right of Fig. 3. Spaxels with velocities above this upper limit are
masked; see the top right plot in Fig. 3 for an example of the resulting mask.
We then select the model with the smallest $\chi^{2}$.

Out of these four models, we select the one with the lowest $\chi^{2}_{\nu}$
that satisfies the requirement $\alpha<100$ as the best-fit model for each
galaxy. An example H$\alpha$ velocity map and best-fit model map is shown in
Fig. 4.

###  3.2 Modeling the stellar mass

Figure 5: Example stellar mass density map extracted from MaNGA Pipe3D (top)
and our best fit to the rotation curve extracted from this map (bottom).

We estimate each galaxy’s stellar mass by fitting a rotation curve due to the
stellar component of the galaxy using the stellar mass density maps available
through the Pipe3D MaNGA analysis pipeline (Sánchez et al., 2016, 2018). An
example stellar mass density map is shown on the top in Fig. 5. Using the
best-fit model H$\alpha$ velocity map values for the galaxy’s kinematic
center, inclination angle, and position angle described above, we define
concentric ellipses that correspond to different orbital radii in the galaxy,
with the radius of each ellipse increasing by 2 spaxels. We compute the
stellar mass as a discretized function of radius, $M_{*}(r)$, by summing the
stellar mass density per spaxel over all the spaxels within each ellipse.

We assume that the stellar mass is the primary component of the galaxy’s disk
and model the stellar mass as the sum of a central bulge and exponential disk.
The rotational velocity due to the bulge and disk is summed in quadrature to
get the rotational velocity due to the stellar mass:

$V_{*}(r)^{2}=V_{b}(r)^{2}+V_{d}(r)^{2}$ (3)

where $V_{*}(r)$ is the rotational velocity due to the stellar mass,
$V_{b}(r)$ is the rotational velocity due to the bulge component, and
$V_{d}(r)$ is the rotational velocity due to the disk component.

The bulge is modeled as an exponential sphere (Sofue, 2017) with rotational
velocity

$V_{b}(r)^{2}=\frac{GM_{0}}{R_{b}}\,F\left(\frac{r}{R_{b}}\right)$ (4)

where $F(x)=1-e^{-x}(1+x+0.5x^{2})$ and $M_{0}=8\,\pi\,R_{b}^{3}\,\rho_{b}$.
The free parameters in this fit are the scale radius of the bulge, $R_{b}$,
and the central density of the bulge, $\rho_{b}$.

The rotational velocity due to the exponential disk (a thin disk without
perturbation; Freeman, 1970) is

$V_{d}(r)^{2}=4\pi G\Sigma_{d}R_{d}y^{2}[I_{0}(y)K_{0}(y)-I_{1}(y)K_{1}(y)]$
(5)

where $\Sigma_{d}$ is the central surface mass density of the disk, $R_{d}$ is
the scale radius of the disk, $y=r/2R_{d}$, and $I_{i}$ and $K_{i}$ are the
modified Bessel functions (Sofue, 2013). The free parameters in this fit are
$\Sigma_{d}$ and $R_{d}$.

##  4 Estimating the mass components

###  4.1 Total mass, $M_{\text{tot}}$

Figure 6: Rotation curves of the 108 MANGA galaxies with H I and H2 masses
from the H$\alpha$ velocity field (top) and the stellar mass component
(bottom). The solid lines extend to the maximum observed distance for each
galaxy, $R_{\text{max}}$, and the dashed lines show the extrapolation of the
model to $R_{90}$. The colors correspond to the different MaNGA samples.

We calculate the galaxy’s total mass dynamical mass within the 90% elliptical
Petrosian radius, $R_{90}$, using the rotational velocity at this radius as
determined from the best-fit rotation curve found as described in Sec. 3.1. We
can calculate the mass of a galaxy within some radius $r$ from the center of
the galaxy under the assumption that the galaxy’s rotational motion is
dominated by Newtonian orbital mechanics. Assuming axial symmetry, the
velocity of a particle at distance $r$ from the center of the galaxy is a
function of the mass within that radius, $M(r)$. Assuming that the orbital
motion is circular in spiral galaxies, the centripetal acceleration of an
orbiting particle is due to the gravitational force,

$M(r)=\frac{V(r)^{2}r}{G}$ (6)

Here, $V(r)$ is the rotational velocity a distance $r$ from the center of the
galaxy, and $G=6.67408\times 10^{-11}$ m3 kg-1 s-2 is the Newtonian
gravitational constant. In order to study the same region of each galaxy, we
estimate the mass within $R_{90}$, $M(R_{90})=\text{$M_{\text{tot}}$}$, by
calculating $V(R_{90})$ from Eqn. 2. When $R_{\text{max}}<R_{90}$, we
extrapolate our parameterization of the fitted rotation curve, Eqn. 2, out to
$R_{90}$. On average, $R_{90}$ exceeds $R_{\text{max}}$ by about 10%. Fig. 6
shows a subset of our rotation curves normalized by $R_{\text{max}}$, where
the curves extrapolated out to $R_{90}$ for the galaxies with
$R_{90}>R_{\text{max}}$ are shown as dashed extensions beyond
$r/R_{\text{max}}=1$. The stellar mass is also evaluated within $R_{90}$, but
only global measurements are available for the remaining mass components.

While the majority of the stellar mass is encompassed by $R_{90}$, gas and
dark matter profiles are known to extend much further than that (e.g.,
Ostriker et al., 1974; Begeman, 1989; Kamphuis & Briggs, 1992; Pohlen et al.,
2010). Extrapolating the rotation curves to higher radii would significantly
increase the uncertainty on the rotational velocity, so we focus our study on
the mass content within the visible extent of the galaxy.

To calculate the total dynamical mass within $R_{90}$, we require:

  * •

$\alpha\leq 99$

  * •

Velocity maps with less than 95% of its spaxels masked

  * •

10 km/s $<V(R_{90})<1000$ km/s

  * •

$\sigma_{V_{\text{max}}}/V_{\text{max}}\leq 2$, where
$\sigma_{V_{\text{max}}}$ is the uncertainty in the best-fit value of
$V_{\text{max}}$.

After applying these quality cuts, our final sample consists of 5522 galaxies
with best-fit rotation curves.

###  4.2 Stellar mass, $M_{*}$

To estimate the total stellar mass of each galaxy, $M_{*}$, within $R_{90}$,
we use the parameters from the best-fit disk and bulge rotation curve as
described in Sec. 3.2. The total mass of the bulge and disk at some radius $r$
is

$M_{*}(r)=M_{b}(r)+M_{d}(r)$ (7)

where the mass of the bulge component within some radius $r$ is

$M_{b}(r)=M_{0}\,F\left(\frac{r}{R_{b}}\right)$ (8)

and the mass of the disk component is

$\displaystyle M_{d}(r)$
$\displaystyle=2\pi\Sigma_{d}\int_{0}^{r}re^{-r/R_{d}}\,dr$ (9)
$\displaystyle=2\pi\Sigma_{d}R_{d}\left[R_{d}-e^{-r/R_{d}}(r+R_{d})\right]$
(10)

So that we study the stellar mass within the same region of each galaxy as the
total mass, we evaluate Eqn. 7 at $R_{90}$.

###  4.3 Atomic hydrogen, H I

We use the H I mass from the H I–MaNGA DR3 survey to quantify the neutral
atomic gas content within each galaxy. As listed in Table 3, H I mass
estimates are available for 2588 galaxies in our sample.

###  4.4 Molecular hydrogen, H2

Figure 7: Top: The dependence of $\log(M_{\text{H}_{2}})$ on $M_{r}$ for 531
galaxies in SDSS DR7 with H2 masses available through CO surveys. The blue
crosses represent blue cloud galaxies, red crosses are green valley and red
sequence galaxies. The points are the mean of the $\log(M_{\text{H}_{2}})$
distribution in each bin in $M_{r}$. The lines are linear fits to the points:
$\log(M_{\text{H}_{2}})$ $=aM_{r}+b$, with the coefficients shown in Table 1.
Bottom: Resolution on $\log(M_{\text{H}_{2}})$: the difference between the H2
mass evaluated based on CO mass and the parameterization from the top plot.
The red line is a fit to a Gaussian with $\sigma=0.27$.

Molecular hydrogen, H2, is a low mass, symmetric molecule without a dipole
moment and therefore does not produce a significant amount of radiation,
making it notoriously difficult to detect. Hence, to evaluate the molecular
hydrogen content in a galaxy, it is customary to parameterize it with respect
to some other observable. The most commonly used method is to use another
molecular gas, particularly CO. We obtain mass estimates of H2 parameterized
by the CO(1-0) line emission from the MASCOT and xCOLD GASS surveys for 108
galaxies that also have total mass, stellar mass, and H I mass estimates (as
described above).

We have CO observations for only a small fraction of our galaxies, so we use
CO observations of SDSS DR7 galaxies to derive a parameterization of the H2
mass as a function of galaxy luminosity in the $r$-band, $M_{r}$. Shown in
Fig. 7, we find the coefficients that describe the linear relationship between
$\log(\text{$M_{\text{H}_{2}}$}/M_{\odot})$ and $M_{r}$:

$\log(\text{$M_{\text{H}_{2}}$}/M_{\odot})=a\,M_{r}+b$ (11)

where $M_{\text{H}_{2}}$ is the mass of molecular hydrogen. The values of $a$
and $b$ are listed in Table 1 and depend on the color-magnitude
classification. We use this parameterization to estimate $M_{\text{H}_{2}}$
when CO observations are not available for galaxies in our sample.

Table 1: $M_{\text{H}_{2}}$ mass parameterization coefficients

CMD classification | $a$ | $b$
---|---|---
Blue cloud | -0.40 ±0.02 | 1.12 ±0.36
Green valley & red sequence | -0.27 ±0.02 | 3.62 ±0.37

Note. — Coefficients for $\log(\text{$M_{\text{H}_{2}}$}/M_{\odot})$
parameterized as a function of $M_{r}$ as shown in Eqn. 11.

###  4.5 Total gas mass, $M_{\text{gas}}$

We define the total gas mass, $M_{\text{gas}}$, as the sum of the H I mass, H2
mass, and helium mass:

$M_{\text{gas}}=\text{$M_{\text{H{\sc
i}}}$}+\text{$M_{\text{H}_{2}}$}+\text{$M_{\text{He}}$}$ (12)

We approximate the helium mass, $M_{\text{He}}$, by assuming a mass fraction
of 25%:

$\text{$M_{\text{He}}$}=\left(\frac{0.25}{1-0.25}\right)(\text{$M_{\text{H{\sc
i}}}$}+\text{$M_{\text{H}_{2}}$})$ (13)

This is the amount of helium measured in the intergalactic medium and agrees
well with the prediction from Big Bang nucleosynthesis (Cooke & Fumagalli,
2018).

###  4.6 Heavy metals and dust mass, $M_{\text{dust}}$

The heavy metal mass and dust mass, $M_{\text{dust}}$, is approximated from a
galaxy’s gas-phase metallicity. We compute the gas-phase metallicity,
$12+\log\left(\frac{\text{O}}{\text{H}}\right)$, following the R-calibration
method described in Pilyugin & Grebel (2016) using the flux of the [O II]
$\lambda\lambda$3727,3729 doublet, [N II] $\lambda$6548, [N II] $\lambda$6584,
[O III] $\lambda$4959, and [O III] $\lambda$5007 emission lines. The fluxes
are extinction-corrected using the Balmer decrement, assuming a flux ratio
H$\alpha$/H$\beta$ $=2.86$. We compute the metallicity as

$\text{$12+\log\left(\frac{\text{O}}{\text{H}}\right)$}=a_{1}+a_{2}\log\left(\frac{R_{3}}{R_{2}}\right)+a_{3}\log
N_{2}\\\ +\left(a_{4}+a_{5}\log\left(\frac{R_{3}}{R_{2}}\right)+a_{6}\log
N_{2}\right)\\\ \times\log R_{2}$ (14)

where

$\displaystyle R_{2}$ $\displaystyle=\frac{\text{[{O~{}II}]}\lambda\lambda
3727,3729}{\text{H$\beta$}}$ (15) $\displaystyle N_{2}$
$\displaystyle=\frac{\text{[{N~{}II}]}\lambda 6548+\text{[{N~{}II}]}\lambda
6584}{\text{H$\beta$}}$ (16) $\displaystyle R_{3}$
$\displaystyle=\frac{\text{[{O~{}III}]}\lambda 4959+\text{[{O~{}III}]}\lambda
5007}{\text{H$\beta$}}$ (17)

are ratios of the specified emission line fluxes. The values of the
coefficients in Eqn. 14 depend on the value of $\log N_{2}$ and are listed in
Table 2.

We assume a constant dust-to-metals ratio corresponding to the metallicity
calibration, $M_{\text{dZ}}$/ $M_{\text{dust}}$ = 0.206 for galaxies with a
gas-phase metallicity greater than 8.2 (De Vis et al., 2019). $M_{\text{dZ}}$
is the dust mass of each galaxy, including metals locked up in dust. The total
mass of heavy metals and dust is then

$M_{\text{dust}}=1.259\,f_{Z}\,M_{g}$ (18)

where $f_{Z}$ is the mass fraction of metals,

$f_{Z}=27.36\left(\frac{\text{O}}{\text{H}}\right)$ (19)

and $M_{g}$ is the gas mass of the galaxy as defined in De Vis et al. (2019):

$M_{g}=\xi\text{$M_{\text{H{\sc
i}}}$}\left(1+\frac{\text{$M_{\text{H}_{2}}$}}{\text{$M_{\text{H{\sc
i}}}$}}\right)$ (20)

where

$\xi=\left(1-(0.2485+1.41f_{Z})-f_{Z}\right)^{-1}$ (21)

Table 2: Gas-phase metallicity coefficients

$\log N_{2}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$
---|---|---|---|---|---|---
≥-0.6 | 8.589 | 0.022 | 0.399 | 0.137 | 0.164 | 0.589
¡ -0.6 | 7.932 | 0.944 | 0.695 | 0.970 | -0.291 | -0.019

Note. — Coefficients for the gas-phase metallicity calculation shown in Eqn.
14, from Pilyugin & Grebel (2016).

###  4.7 The total visible mass, $M_{\text{vis}}$

Figure 8: The relative contributions of each mass component to the total
visible mass of SDSS DR7 galaxies as a function of $M_{r}$. For simplicity, we
only show $M_{\text{H}_{2}}$ parameterized as a function of $M_{r}$ here.

We define the total visible mass of a galaxy, $M_{\text{vis}}$, as the sum of
the stellar mass, $M_{*}$, the gas mass, $M_{\text{gas}}$ (Eqn. 12), and the
heavy metals and dust mass, $M_{\text{dust}}$ (Eqn. 18):

$\text{$M_{\text{vis}}$}=M_{*}+\text{$M_{\text{H{\sc
i}}}$}+\text{$M_{\text{H}_{2}}$}+\text{$M_{\text{He}}$}+\text{$M_{\text{dust}}$}$
(22)

A summary of the relative contributions of each individual mass component to
the total visible mass for SDSS DR7 galaxies, as a function of the $r$-band
luminosity, $M_{r}$, is shown in Fig. 8. For galaxies with $M_{r}>-18$, gas is
the dominant component of the visible mass, whereas for galaxies with
$M_{r}<-19$, the stellar mass dominates the visible mass. Heavy metals and
dust contribute on the order of 1% regardless of magnitude.

##  5 Statistically modeling the rotational velocity

Figure 9: Illustration of the statistical model. Red horizontal arrows denote
Gaussian smearing with the corresponding $\sigma$.

To test our null hypothesis, i.e. galaxies do not have a dark matter halo,
hence the observed rotational velocity at $R_{90}$ is due entirely to visible
mass, we construct a statistical model to predict the expected rotational
velocity of a galaxy given its total visible mass. We choose the ratio of the
expected to observed velocity evaluated at $R_{90}$,
$V_{\text{exp}}/V_{\text{obs}}$, as our observable. The expected velocity is
evaluated based on the visible mass calculated using the methods described
below. A value of this observable close to unity signals consistency of the
data with the null hypothesis.

The resolution of this observable is determined by the measured uncertainty of
each visible mass component and the uncertainty of the fitted rotation curve
to the velocity map, from which we determine the velocity at $R_{90}$. We
expect the velocity to be normally distributed around its true value with the
uncertainty returned by the fit.

To evaluate the effect of these uncertainties, we implement the following
procedure. First, for each galaxy, we determine the mass of each component of
the visible mass as described in Sec. 4.2–4.6. Since $M_{\text{H}_{2}}$ is
available from CO observations for only a small number of galaxies, we also
use the parameterization as a function of $M_{r}$ to estimate
$M_{\text{H}_{2}}$ described in Sec. 4.4. We estimate the total mass,
$M_{\text{tot}}$, from the best-fit rotation curve as described in Sec. 4.1.
We then compute the ratio of visible to total mass,

$F_{\text{vis}}=\frac{\text{$M_{\text{vis}}$}}{\text{$M_{\text{tot}}$}}$ (23)

for each galaxy.

To statistically determine the rotational velocity, we smear each mass
component according to its expected resolution111Since we observe a Gaussian
distribution in $\log M$ of the corresponding component, we randomly smear
$\log M$ according to a Gaussian distribution and then invert to find the
corresponding mass.. The expected velocity, $V_{\text{exp}}$, is then
evaluated based on the sum of each of these smeared mass components and is
smeared according to the velocity uncertainty from the fit to the rotation
curve. This smearing procedure is repeated 1000 times for each galaxy. A
schematic of this statistical model is illustrated in Fig. 9. From this
procedure, we find the expected fraction of the instances where the observed
rotational velocity is less than the rotational velocity expected from just
the visible mass components, $F(V_{\text{obs}}<V_{\text{exp}})$, where
$V_{\text{obs}}$ is the rotational velocity measured at $R_{90}$ from the
best-fit rotation curve.

##  6 Studying the ratio of visible to total mass

width=0.49Fig10a.eps [image label = $M_{\text{vis}}=M_{*}$ at north];
width=0.49Fig10b.eps [image label = $M_{\text{vis}}=M_{*}$ \+ $M_{\text{H{\sc
i}}}$ \+ $M_{\text{H}_{2}}(M_{r})$ \+ $M_{\text{He}}$ at north];
width=0.49Fig10c.eps [image label = $M_{\text{vis}}=M_{*}$ \+ $M_{\text{H{\sc
i}}}$ \+ $M_{\text{H}_{2}}(\text{CO})$ \+ $M_{\text{He}}$ at north];

Figure 10: The PDF of the ratio of expected to observed velocities at
$R_{90}$. The black points are the data with the expected velocity evaluated
from the visible mass without smearing. The colored histograms are the PDF
evaluated based on the statistical model for a sample of randomly selected
galaxies. The red histogram is the normalized sum of the individual PDFs. The
vertical black line at 1 corresponds to the observed and expected velocities
being equal. The integral of the PDFs to the right of this line corresponds to
the observed (black points) and modeled (red histogram)
$F(V_{\text{obs}}<V_{\text{exp}})$ listed in Table 3. Top row: Only stellar
mass contributes to the visible mass. Second row: Gas mas is added to stellar
mass, with $M_{\text{H}_{2}}$ determined from $M_{r}$. Third row: Same as the
second row, but $M_{\text{H}_{2}}$ is determined from CO observations.

Figure 11: The dependence of various mass fractions on luminosity for blue
cloud galaxies (top) and green valley and red sequence galaxies (bottom). The
purple circles compare just the stellar mass to total mass, the cyan triangles
compare the visible mass (with $M_{\text{H}_{2}}$ estimated from $M_{r}$) to
total mass, and the black triangles compare the visible mass (with
$M_{\text{H}_{2}}$ inferred from CO observations) to total mass. The black
line at 1 is where the visible mass is equal to the total mass.

In Fig. 10, we present the PDFs of the ratio of expected to observed
velocities at $R_{90}$ derived using the statistical model and the
distribution observed in data. The integrals of these distributions above 1
correspond to the fractions of galaxies for which the expected velocity
exceeds the observed one, $F(V_{\text{obs}}<V_{\text{exp}})$, listed in Table
3. In Table 3 we also present the mean and RMS of $F_{\text{vis}}$ (the ratio
of visible to total mass, as described in Sec. 5).

We break down the sample into a number of different subsets: by CMD class into
blue cloud, green valley, and red sequence; and by MaNGA targeting sample (to
check for possible systematic bias). Due to the limited statistics, we combine
galaxies in the green valley and red sequence. For each sample of galaxies, we
consider three different mass ratios: $M_{*}/$$M_{\text{tot}}$ (labeled “Only
stars” in Table 3), $M_{\text{vis}}$/$M_{\text{tot}}$ with $M_{\text{H}_{2}}$
inferred from $M_{r}$, and $M_{\text{vis}}$/$M_{\text{tot}}$ with
$M_{\text{H}_{2}}$ measured with CO observations.

Across all galaxy samples and subsamples, the null hypothesis for the ratio
$M_{*}/$$M_{\text{tot}}$ is clearly inconsistent with the data. The preferred
value of $F_{\text{vis}}$ is 40–50% for all of the galaxy samples. When we
include all visible mass, with $M_{\text{H}_{2}}$ parameterized by $M_{r}$,
$F_{\text{vis}}$ increases to 50–55%, reducing the variation between the
different categories. The effect is more dramatic for galaxies in the blue
cloud, where gas contributes a significant amount the mass budget. Finally,
when we use $M_{\text{H}_{2}}$ estimated from CO observations, which is a more
reliable method than our parameterization with $M_{r}$, we again see
$F_{\text{vis}}$ increase to $\sim$65%. As we go from just $M_{*}$ to
$M_{\text{vis}}$ with $M_{\text{H}_{2}}$ estimated from CO, we see a continual
increase in the fraction of galaxies with $V_{\text{obs}}<V_{\text{exp}}$. As
shown by the values in Table 3, we find no statistically significant
difference between the three MaNGA targeting samples.

The remaining component of the baryonic mass that is missing from our analysis
is ionized hydrogen, H II. We expect the H II mass to be on the same order of
magnitude as $M_{\text{He}}$, with star-forming galaxies containing more H II,
but we do not anticipate that the inclusion of H II to significantly change
our results.

Finally, we show the dependence of $F_{\text{vis}}$ on luminosity for galaxies
in the blue cloud, green valley and red sequence in Fig. 11. When only stellar
mass is included in the visible mass estimation, the dependence of
$F_{\text{vis}}$ on $M_{r}$ is rather flat for green valley and red sequence
galaxies, while for the blue cloud galaxies there is a notable upward trend,
with brighter galaxies having a larger ratio of $M_{*}$/$M_{\text{tot}}$. This
matches results from previous studies of the stellar-halo mass relation,
including Persic et al. (1996); Strigari et al. (2008); Torres-Flores et al.
(2011); Karukes & Salucci (2017); Behroozi et al. (2019); Di Paolo et al.
(2019); Douglass et al. (2019); Douglass & Demina (2022), and from the
simulations by Moster et al. (2010). We find, though, that once gas is added
to the visible mass, the dependence of $F_{\text{vis}}$ on $M_{r}$ becomes
rather flat for blue cloud galaxies as well. Though lower in statistics, the
distribution for galaxies with $M_{\text{H}_{2}}$ evaluated based on CO
confirms this behavior. Based on this observation, we conclude that the dark
matter content is independent of galaxy luminosity or evolutionary stage. This
conclusion contradicts a widespread belief that dwarf galaxies are rich in
dark matter (Pryor & Kormendy, 1990; Binney & Tremaine, 2008). Yet, as shown
in Fig. 8, the visible mass in such galaxies is dominated by gas, which should
be properly accounted for when dark matter content is evaluated.

Table 3: Mass ratio statistics for MaNGA DR17 galaxies.

|  | $F(V_{\text{obs}}<V_{\text{exp}})$ |  $F_{\text{vis}}$
---|---|---|---
Sample | Count | Observed | Modeled | Mean | RMS
Only stars
All | 5522 | 5.4% | 5.7% | 45±0.5% | 35±0.4%
Blue cloud | 3025 | 3.3% | 3.7% | 40±0.6% | 30±0.4%
Green valley, red sequence | 1945 | 8.2% | 8.6% | 52±0.9% | 39±0.7%
MaNGA sample 1 | 2476 | 4.8% | 5,3% | 45±0.8% | 35±0.6%
MaNGA sample 2 | 2073 | 5.5% | 5.5% | 44±0.8% | 33±0.6%
MaNGA sample 3 | 943 | 6.5% | 6.9% | 46 ±1.3% | 37±0.9%
Stars, dust, H I, H2($M_{r}$), He
All | 2588 | 9.1% | 9.8% | 52±0.8% | 38±0.6%
Blue cloud | 1743 | 8.6% | 9.4% | 51±0.9% | 37±0.7%
Green valley, red sequence | 560 | 9.8% | 10.3% | 55±1.9% | 41±1.3%
MaNGA sample 1 | 1588 | 9.0% | 9.7% | 52±1.0 % | 39±0.7%
MaNGA sample 2 | 560 | 7.7% | 8.4% | 51±1.5% | 34±1.0%
MaNGA sample 3 | 430 | 10.7% | 11.2% | 53±2.0% | 40±1.4%
Stars, dust, H I, H2(CO), He
All | 108 | 18.5% | 17.0% | 65±5% | 49±4%
Blue cloud | 75 | 21% | 19% | 66±7% | 54±5%
Green valley, red sequence | 28 | 14% | 16% | 64±7% | 37±5%

Note. — The observed velocity, $V_{\text{obs}}$, is evaluated at $R_{90}$
based on the fit to the rotation curve. The expected velocity,
$V_{\text{exp}}$, is evaluated based on the visible mass.
$F(V_{\text{obs}}<V_{\text{exp}})$ is the fraction of galaxies for which
$V_{\text{obs}}<V_{\text{exp}}$. In the “Modeled” column, the visible mass and
$V_{\text{exp}}$ are distributed according to the statistical model; in the
“Observed” column, they are not smeared. $F_{\text{vis}}$ is the fraction of
the visible mass, i.e. the ratio of the visible to total mass. Color
classification and MaNGA sample information may not be available for all
galaxies.

###  6.1 Comparison to previous results

As shown in Fig. 11, we find that, once we account for all of the visible mass
components of a galaxy, the ratio of $M_{\text{vis}}$/$M_{\text{tot}}$ does
not depend on a galaxy’s luminosity or evolutionary stage. This is contrary to
previous work by Torres-Flores et al. (2011), who consider the relationship
between $M_{\text{vis}}$, defined as stellar mass and H I mass, and total
mass. Torres-Flores et al. (2011) find a correlation between the mass ratio
and evolutionary stage in that late-type low mass spirals are dominated by
dark matter in comparison to early-type high mass spirals. This discrepancy
could be attributed to our inclusion of both H2 and helium mass, which
contribute more to the visible mass of fainter galaxies.

$M_{\text{vis}}$/$M_{\text{tot}}$ is a version of the stellar-halo mass
relation (SHMR) typically described as the ratio of stellar mass to halo mass.
Models predict a SHMR that deviates from a flat distribution (e.g. Behroozi et
al., 2019), with lower values for the faintest and brightest galaxies. These
galaxies are thought to be dominated by dark matter. Similar to Douglass &
Demina (2022), we find that on the faint end of blue cloud galaxies, the
inclusion of gas significantly increases $M_{\text{vis}}$/$M_{\text{tot}}$. We
find that these galaxies are not dominated by dark matter, but instead have
the same amount of dark matter relative to their visible mass as brighter
galaxies, suggesting that the lower stellar mass in these galaxies is a result
of visible matter that is still in the gas phase. A flat SHMR indicates that
halo size is strongly related to galaxy size and plays an important role in
galaxy evolution.

##  7 Conclusions

We study the ratio of visible to total mass in spiral galaxies using rotation
curves evaluated with the H$\alpha$ velocity maps from SDSS MaNGA DR17. From
the dependence of the rotational velocity on the distance from the center of a
galaxy, we evaluate the velocity at the 90% elliptical Petrosian radius,
$R_{90}$ from the fitted rotation curves. We compute the visible mass of each
galaxy, which includes stellar mass evaluated at the same radius $R_{90}$, the
mass of atomic hydrogen (H I), molecular hydrogen (H2) evaluated based on the
CO content, helium, and heavy metals and dust mass. To increase the size of
the sample under study, we also use a parameterization of $M_{\text{H}_{2}}$
as a function of the galaxy luminosity in the $r$-band, $M_{r}$, derived using
the SDSS DR7 galaxy sample. The helium mass is added assuming that its mass
fraction in the total gas amount is 25%.

We construct a statistical model that predicts the velocity based on the
visible mass and compares it to the observed velocity. If the expected
velocity is evaluated based solely on the stellar mass, the expected velocity
exceeds the observed velocity in only 3–9% of the cases. After including all
of the gas and dust mass, this fraction increases to 8–20%, depending on the
sample selection and method for estimating $M_{\text{H}_{2}}$. Hence, the null
hypothesis (no dark matter) cannot be excluded at a confidence level better
than 95% for the mass within the visible extent of disk galaxies. However, we
do observe that, once all of the visible mass is accounted for, the ratio of
visible to total mass is largely independent of a galaxy’s luminosity or
evolutionary stage.

Future work will incorporate the mass of ionized hydrogen and extend the mass
component analysis to elliptical galaxies.

## Acknowledgements

The authors would like to thank Bob Cousins for insightful remarks on the
statistical model, and Eric Blackman and Alice Quillen for careful reading and
thoughtful comments.

N.R. acknowledges support from the Feinberg Research Award through the
Department of Physics & Astronomy at the University of Rochester. R.D.
acknowledges support from the Department of Energy under the grant DE-
SC0008475.0.

This project makes use of the MaNGA-Pipe3D data products. We thank the IA-UNAM
MaNGA team for creating this catalogue, and the Conacyt Project CB-285080 for
supporting them.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

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# The Hat Guessing Number of Cactus Graphs and Cycles

Jeremy Chizewer Corresponding Author. Department of Combinatorics and
Optimization, University of Waterloo, Email<EMAIL_ADDRESS> I.M.J.
McInnis Department of Mathematics, Princeton University, Email:
<EMAIL_ADDRESS>Mehrdad Sohrabi  Department of Combinatorics and
Optimzation and Department of Computer Science, University of Waterloo, Email:
<EMAIL_ADDRESS>Shriya Kaistha Department of Computer Science,
University of Waterloo, Email<EMAIL_ADDRESS>
###### Abstract

We study the hat guessing game on graphs. In this game, a player is placed on
each vertex $v$ of a graph $G$ and assigned a colored hat from $h(v)$ possible
colors. Each player makes a deterministic guess on their hat color based on
the colors assigned to the players on neighboring vertices, and the players
win if at least one player correctly guesses his assigned color. If there
exists a strategy that ensures at least one player guesses correctly for every
possible assignment of colors, the game defined by $\langle G,h\rangle$ is
called winning. The hat guessing number of $G$ is the largest integer $q$ so
that if $h(v)=q$ for all $v\in G$ then $\langle G,h\rangle$ is winning.

In this note, we determine whether $\langle G,h\rangle$ is winning for any $h$
whenever $G$ is a cycle, resolving a conjecture of Kokhas and Latyshev in the
affirmative, and extending it. We then use this result to determine the hat
guessing number of every cactus graph, graphs in which every pair of cycles
share at most one vertex.

##  1 Introduction

Given a simple undirected graph $G$ and a function $h:V(G)\to\mathbb{N}$
mapping the vertices of $G$ to the positive integers, the _hat guessing game_
denoted $\langle G,h\rangle$ is a puzzle in which for every vertex $v\in G$ a
player is placed and assigned a hat from $h(v)$ possible colors. Each player
attempts to guess the color of his hat according to a predetermined strategy
that takes as input the colors assigned to players at neighboring vertices.
The players are a team and are said to _win_ for a given coloring if at least
one player guesses correctly. The hat guessing game $\langle G,h\rangle$ is
said to be _winning_ if there exists some set of strategies for the players so
that for every possible color assignment, the players win. The players _lose_
on a coloring if no player guesses correctly; if there exists such a coloring
for every possible set of guessing strategies, the game is called _losing_.

We refer to the function $h$ as the _hatness_ function, and for a vertex $v\in
G$ we call the value $h(v)$ the hatness of $v$. A _subgame_ $\langle
G^{\prime},h^{\prime}\rangle$ of a hat guessing game $\langle G,h\rangle$ is a
game where $G^{\prime}\subseteq G$ is a subgraph, and the hatness function
$h^{\prime}=h|_{v\in G^{\prime}}$ is the hatness function $h$ restricted to
the vertices of $G$ in $G^{\prime}$. Moreover, we call a subgame _proper_ if
$G^{\prime}$ is a proper subgraph of $G$. The _hat guessing number_ of a
graph, $G$, denoted $HG(G)$, is the largest integer $q$ such that $\langle
G,\star q\rangle$ is a winning game, where $\star q$ denotes the constant
function $h(\cdot)=q$. We call a vertex $v$ _deletable_ if the subgame on
$G\backslash\\{v\\}$ is winning. Note that if a game has a winning subgame
then it is winning, and if a game is winning then decreasing the hatness of
any vertex results in a winning game.

The hat guessing number was introduced in [4]. Previous results have attempted
to bound the hat guessing number of graphs, but few give exact answers. Some
exact results include the hat guessing number of trees (folklore), cycles
[15], pseudotrees [10], complete graphs (folklore), and some windmills and
book graphs [7]. There are also upper and lower bounds (that are somewhat far
apart) on complete bipartite graphs [4, 6, 1], upper bounds on outerplanar
graphs [3, 9], and lower bounds on planar graphs [2, 12, 14].

In this note, we study the hat guessing problem for cycles and cactus graphs.
Our first result resolves [11, Conjecture 5.2] affirmatively and goes further,
showing exactly which games on cycles are winning. The second result adds
cactus graphs to the growing list of graph classes for which the hat guessing
number is known exactly.

######  Theorem 1.1.

Let $C$ be a cycle on $n$ vertices such that the hatness $h$ of each vertex
$v$ in $C$ satisfies $h(v)\geq 2$. Then $\langle C,h\rangle$ is winning if and
only if $C$ satisfies at least one of the following conditions:

  1. 1.

The length is $4$ or divisible by $3$ and $h(v)\leq 3$ for all $v\in C$.

  2. 2.

The length is $3$ and $\sum_{v\in C}\frac{1}{h(v)}\geq 1$.

  3. 3.

The game $\langle C,h\rangle$ contains a winning proper subgame.

  4. 4.

The hatnesses satisfy both of the following properties:

    1. (a)

There exists a sequence of adjacent vertices in $C$ with hatnesses $(2,3,3)$
or $(3,2,3)$.

    2. (b)

For all $v\in C$, $h(v)\leq 4$,

The forward implication for Condition 1 is [15, Theorem 1]; for Condition 2,
it is [11, Theorem 2.1]; for Condition 3, it is trivial; and for Condition 4,
it is [11, Theorem 5.1]. We prove the reverse implications—that $\langle
C,h\rangle$ is losing if none of the above conditions hold—in Section 2.

Our next result determines the hat guessing number of cactus graphs. A _cactus
graph_ is a connected graph in which every pair of cycles share at most one
vertex. Figure 1 shows an example of a cactus graph (on the left). A useful
subset of the class of cactus graphs is the class of _pseudotrees_ , which are
connected graphs with at most one cycle. The hat guessing number of cactus
graphs was first studied in [3],111The bound of $16$ was mentioned in the
preprint (arXiv:2109.13422) but was removed from the publication which derived
a bound of $16$ for any cactus graph. We apply Theorem 1.1 to exactly
determine the hat guessing number of every cactus graph.

######  Theorem 1.2.

Let $G$ be a cactus graph.

  1. 1.

$HG(G)=4$ if and only if $G$ contains at least two triangles.

  2. 2.

$HG(G)=3$ if and only if $G$ contains at least two cycles or a cycle of length
$4$ or divisible by $3$, and $G$ contains fewer than two triangles.

  3. 3.

$HG(G)=2$ if and only if $G$ is a pseudotree with at least one edge and no
cycle of length $4$ or divisible by $3$.

The lower bound for Statement 1 of the theorem follows from Theorem 2.1 by
gluing together two triangles, each with hatnesses $(2,4,4)$ to either end of
a path (possibly of length 0) with hatnesses $(2,4,\dots,4,2)$ at the vertices
with hatness $2$ (as shown in Figure 1 on the right). Since this forms a
winning subgame of $\langle G,\star 4\rangle$, the game $\langle G,\star
4\rangle$ is winning. The lower bound for Statement 2 in the two cycles case
follows analogously by gluing two winning cycles with hatnesses
$(2,3,\dots,3)$ to either end of a winning path (possibly of length 0) with
hatnesses $(2,3,\dots,3,2)$ and decreasing all hatnesses in the resulting game
to $3$ to give a winning subgame of $\langle G,\star 3\rangle$. The lower
bound for Statement 2 in the one cycle case follows from [15, Theorem 1]. The
lower bound for Statement 3 is trivial. In Section 3, we prove the upper
bounds.

${4}$${4}$${2}$${2}$${4}$${4}$${4}$${2}$${2}$${4}$

Figure 1: A cactus graph with two triangles (left) and the construction that
shows that the hat guessing number is at least $4$ (right). The triple lines
indicate vertices that are glued together in the construction.

##  2 Cycles

Throughout this section and the next we use _constructors_ —rules for building
winning or losing games from smaller games that are easier to analyze—to
simplify our proofs. Given two graphs $G_{1}$ and $G_{2}$, with vertices
$v_{1}$ and $v_{2}$, respectively, the graph denoted
$G=G_{1}+_{v_{1},v_{2}}G_{2}$ formed by _gluing_ $G_{1}$ and $G_{2}$ at
$v_{1}$ and $v_{2}$ is given by the union of vertices and edges $G=G_{1}\cup
G_{2}$ where $v_{1}$ and $v_{2}$ are identified as a single vertex in $G$
which is incident to all vertices in the union $N_{G_{1}}(v_{1})\cup
N_{G_{2}}(v_{2})$ of their respective neighbor sets in the original graphs.
The following theorems, which concern the gluing of winning and losing graphs,
are used as building blocks to prove our main results.

######  Theorem 2.1 ([11, Theorem 3.1]).

Let $G_{1}$ and $G_{2}$ be graphs with $v_{1}\in G_{1}$ and $v_{2}\in G_{2}$
and let $G=G_{1}+_{v_{1},v_{2}}G_{2}$. Let $h_{1}$ and $h_{2}$ be hatness
functions for the graphs $G_{1}$ and $G_{2}$ respectively, and suppose that
$\langle G_{1},h_{1}\rangle$ and $\langle G_{2},h_{2}\rangle$ are both
winning. Then $\langle G,h\rangle$ is winning where

$h(v)=\begin{cases}h_{1}(v)&\text{ if }v\in G_{1}\backslash\\{v_{1}\\}\\\
h_{2}(v)&\text{ if }v\in G_{2}\backslash\\{v_{2}\\}\\\ h_{1}(v_{1})\cdot
h_{2}(v_{2})&\text{ if }v=v_{1}=v_{2}\\\ \end{cases}$

######  Theorem 2.2 ([13, Theorem 4.1]).

Let $G_{1}$ and $G_{2}$ be graphs with $v_{1}\in G_{1}$ and $v_{2}\in G_{2}$
and let $G=G_{1}+_{v_{1},v_{2}}G_{2}$. Let $h_{1}$ and $h_{2}$ be hatness
functions for the graphs $G_{1}$ and $G_{2}$ respectively, and suppose that
$\langle G_{1},h_{1}\rangle$ and $\langle G_{2},h_{2}\rangle$ are both losing
with $h_{1}(v_{1})\geq h_{2}(v_{2})=2$. Then $\langle G,h\rangle$ is losing
where

$h(v)=\begin{cases}h_{1}(v)&\text{ if }v\in G_{1}\\\ h_{2}(v)&\text{ if }v\in
G_{2}\backslash\\{v_{2}\\}\end{cases}$

With these results in mind, we are ready to move on to the proof of the
reverse implications in Theorem 1.1. To prove this direction, we must show
that for a game $\langle C,h\rangle$ when none of the conditions hold, the
game is losing. If none of the conditions hold and $n=3$, the result follows
from [11, Theorem 2.1]. Hence, we may assume $n>3$ for the rest of the proof.
Further, if there are no vertices of hatness $2$, the result holds by [15,
Theorem 1], so we may assume for the rest of the proof that there is at least
one vertex $a$ with $h(a)=2$. We split the proof into two parts: Condition 4
fails because Condition 4a fails, and Condition 4 fails because Condition 4b
fails.

###  2.1 Proof of Theorem 1.1 when Condition 4a Fails

###### Proof.

We begin by assuming that Conditions 1, 2, 3, and 4a all fail, and Condition
4b holds. Given these assumptions, let $C$ be a cycle of length $n>3$ such
that for every $v\in C$ we have $2\leq h(v)\leq 4$, and let $h(a)=2$ for some
vertex $a\in C$. As a reminder, the assumptions also imply $\langle
C,h\rangle$ does not contain a winning subgame and the hatness sequences
$(3,2,3)$ and $(2,3,3)$ do not appear. Note that $a$ is the only vertex with
hatness $2$ as otherwise Condition 3 would be satisfied by repeated
application of Theorem 2.1 to the game $\langle K_{2},\star 2\rangle$.

Based on the supposition, we have two possible minimal cases, as suggested in
[11]. It suffices to prove the claim for these minimal cases as increasing the
hatnesses of any vertex cannot cause a losing game to become winning. The
first case is that the hatnesses of $a$’s neighbors are each $4$, and every
other hatness is $3$. The second case is that $a$ has one neighbor $b$ with
hatness $3$ and one with hatness $4$, and the second neighbor of $b$ has
hatness $4$. Any other vertices in the cycle have hatness $3$ if they exist.
We prove the result for these two possibilities separately, although the
technique is quite similar.

$v$$a$$b$$c$$\,\dots$$4$ $2$ $4$ $3$ $3$

$v$$a$$b$$c$$\,\dots$$4$ $2$ $3$ $4$ $3$

Figure 2: The two cases described in the proof of Theorem 1.1, where the
ellipsis indicates an arbitrary (possibly 0) number of vertices with hatness
$3$.

In the first case, let $v,a,b,c$ be a sequence of vertices in the cycle with

$h(v)=4,\,h(a)=2,\,h(b)=4,\,h(c)=3$

and any other vertices have hatness 3, as shown on the left side of Figure 2.

$c$$\,\dots$$v$$a$$b$$3$ $2$ $3$

$c$$\,\dots$$v$$a$$b$$2$ $3$ $3$

Figure 3: The two cases described in the proof of Theorem 1.1, after coloring
$a$ and $b$ and committing to a subset of colors for $c$ and $v$. The ellipsis
indicates an arbitrary (possibly 0) number of vertices with hatness $3$.

Let $x_{b}$ be the color guessed least frequently by vertex $b$ (breaking ties
arbitrarily). By the pigeonhole principle, $b$ can only guess $x_{b}$ at most
once. Let $x_{a}$ and $x_{c}$ be the colors of $a$ and $c$, respectively, when
$b$ guesses $x_{b}$ (if $x_{b}$ is never guessed, $a$ and $c$ can be chosen
arbitrarily). Let $X_{v}$ be the set of colors for $v$ so that when $b$ is
colored $x_{b}$, and $v$ is colored from $X_{v}$, $a$ guesses color $x_{a}$.
Suppose toward a contradiction that $|X_{v}|\geq 2$. Then we commit to
coloring $a$ with the color which is not $x_{a}$, $b$ with the color $x_{b}$,
$c$ with any of the three colors, and $v$ with a color from $X_{v}$, so $a$
and $b$ guess incorrectly. Hence, $\langle C,h\rangle$ is winning only if
there exists a winning strategy for the game on a path of length at least $2$
where every vertex has hatness $3$ except for one vertex with hatness $2$ as
depicted in the right side of Figure 3. However, Theorem 2.2 implies that this
path is losing. Hence, in any winning strategy, we may assume that
$|X_{v}|\leq 1$. Thus, we can commit to coloring $a$ with $x_{a}$, $v$ with
one of at least $3$ colors not in $X_{v}$, coloring $b$ with $x_{b}$ and $c$
with one of the two colors not equal to $x_{c}$. This ensures again that $a$
and $b$ each guess incorrectly, and we again reduce to the path where every
vertex has hatness $3$ except one with hatness $2$ (in this case depicted on
the left side of Figure 3), which is losing. Hence, $C$ is losing.

Next, we prove the result in the second case, where we have a sequence of
vertices $v,a,b,c$ in the cycle with

$h(v)=4,\,h(a)=2,\,h(b)=3,\,h(c)=4,$

and any other vertices in the cycle have hatness $3$ if they exist (as shown
on the right side of Figure 2). As before, we let $x_{b}$ be the least
frequently guessed color for $b$, breaking ties arbitrarily. By the pigeonhole
principle, $x_{b}$ is guessed at most twice. Let $(x_{a},x_{c})$ and
$(y_{a},y_{c})$ be the colorings of $(a,c)$ for which $b$ guesses $x_{b}$. If
$x_{b}$ is guessed at most once then we can choose arbitrary values for any
undetermined pairs. Now, there are three possibilities to consider. The first
case is $x_{c}=y_{c}$; the second is $x_{a}=y_{a}$; the third is $x_{a}\neq
y_{a}$ and $x_{c}\neq y_{c}$.

In the first case, we commit to coloring $b$ with the color $x_{b}$, and $c$
with a color other than $x_{c}$. This reduces to a path where the first vertex
has hatness $2$, the second vertex has hatness $4$, and the remaining vertices
have hatness $3$, which is losing by Theorem 2.2.

In the second case, we proceed as in the $(4,2,4,3)$ case. Let $X_{v}$ be the
set of colors for $v$ such that $a$ guesses $x_{a}$. As before, if
$|X_{v}|\geq 2$ we color $a$ with the color not equal to $x_{a}$, $v$ from
$X_{v}$ and $c$ arbitrarily; otherwise, we color $a$ with $x_{a}$, $v$ from
colors not in $X_{v}$ and $c$ from colors not $x_{c}$ or $y_{c}$. As before,
these choices ensure that $a$ and $b$ guess incorrectly and leave losing
paths. (Shown on the right and left sides of Figure 3, respectively. We
actually have $4$ possible colors for $c$ in the first case but only $3$ are
needed.)

Finally, in the third case, assume without loss of generality that $a$ guesses
$x_{a}$ at most as frequently as $y_{a}$ when $b$ is colored $x_{b}$. Then we
can commit to coloring $b$ with the color $x_{b}$, vertex $a$ with the color
$x_{a}$, and $c$ with a color that is not $x_{c}$, of which there are $3$
choices. Since $x_{a}$ is guessed at most as frequently as $y_{a}$, there are
at least $2$ colors for $v$ such that $a$ does not guess $x_{a}$, and we can
again reduce the game to a losing path (in this case, the one shown on the
right in Figure 3). Thus, $\langle C,h\rangle$ is losing in all cases when
Conditions 1, 2, 3, and 4a fail.∎

It remains to show that $\langle C,h\rangle$ is losing when Conditions 1, 2,
3, and 4b fail.

###  2.2 Proof of Theorem 1.1 when Condition 4b Fails

We begin with lemmas to simplify the argument for this case. Our first lemma
allows us to delete two vertices of a cycle, leaving behind a path that is
winning if the original cycle is winning. An application of the lemma is
depicted in Figure 4.

######  Lemma 2.3.

Let $C=(V,E)$ be a cycle of length at least $4$, let $t,u,v,w\in V$ be
consecutive vertices in the cycle so that $(t,u),(u,v),(v,w)\in E$, and let
$h(\cdot)$ be a hatness function with $h(v)>h(u)$. We define

$h^{\prime}(x)=\begin{cases}\left\lceil
h(w)\left(1-\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)\right\rceil&\text{if
}x=w\\\ h(t)-\left\lfloor\frac{h(t)}{h(u)}\right\rfloor&\text{if }x=t\\\
h(x)&\text{if }x\in V\backslash\\{t,u,v,w\\}\end{cases}$

If $\langle C,h\rangle$ is winning then $\langle
C\backslash\\{u,v\\},h^{\prime}\rangle$ is winning.

$t$$a_{t}$ $u$$a_{u}$ $v$$a_{v}$ $w$$a_{w}$ $x$$a_{x}$ $y$$a_{y}$ $\left\lceil
a_{w}\left(1-\left\lceil\frac{a_{v}}{a_{u}}\right\rceil^{-1}\right)\right\rceil$$a_{t}-\left\lfloor\frac{a_{t}}{a_{u}}\right\rfloor$

$w$$x$$y$$t$$u$$v$$a_{t}-\left\lfloor\frac{a_{t}}{a_{u}}\right\rfloor$ $a_{y}$
$a_{x}$ $\left\lceil
a_{w}\left(1-\left\lceil\frac{a_{v}}{a_{u}}\right\rceil^{-1}\right)\right\rceil$

Figure 4: A cycle before and after applying Lemma 2.3

###### Proof.

The first step is to show that there exists a color $x_{v}\in[h(v)]$ such that
for each color $x_{u}\in[h(u)]$ there is a set
$X_{w}(x_{u},x_{v})\subseteq[h(w)]$ of colors for $w$ with

$|X_{w}(x_{u},x_{v})|\geq\left\lceil
h(w)\left(1-\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)\right\rceil,$
(1)

so that if $v$ is colored $x_{v}$, $u$ is colored $x_{u}$, and $w$ is colored
$x_{w}\in X_{w}(x_{u},x_{v})$ then $v$ guesses incorrectly. Indeed, suppose
toward a contradiction that no such $x_{v}$ exists. Then, for each color
$x_{v}\in[h(v)]$, there exists a color $x_{u}\in[h(u)]$ that violates Equation
1. That is, there exists a set $Y_{w}(x_{u},x_{v})\subseteq[h(w)]$ of colors
for $w$ such that $v$ correctly guesses $x_{v}$ whenever $w$ is colored from
$Y_{w}(x_{u},x_{v})$ and $u$ is colored $x_{u}$, and

$|Y_{w}(x_{u},x_{v})|>h(w)-\left\lceil
h(w)\left(1-\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)\right\rceil.$

By the pigeonhole principle, there is some $y_{u}\in[h(u)]$ which is used to
generate $Y_{w}(y_{u},x_{v})$ for a subset $Y_{v}\subseteq[h(v)]$ of colors
with $|Y_{v}|\geq\left\lceil\frac{h(v)}{h(u)}\right\rceil$. Summing over this
set, we get

$\displaystyle\sum_{y_{v}\in Y_{v}}|Y_{w}(y_{u},y_{v})|$
$\displaystyle>\left\lceil\frac{h(v)}{h(u)}\right\rceil\left(h(w)-\left\lceil
h(w)\left(1-\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)\right\rceil\right)$
$\displaystyle\geq\left\lceil\frac{h(v)}{h(u)}\right\rceil\left(h(w)\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)$
$\displaystyle=h(w).$

This is a contradiction since for a fixed color $y_{u}$, there are exactly
$h(w)$ guesses made by $v$. Hence, there exists some $z_{v}\in[h(v)]$ with the
desired property.

Fix the color of $v$ to be $z_{v}$. With $z_{v}$ fixed, fix the color of $u$
to be the least frequently guessed color of $u$, denoted $z_{u}$. By our
choice of $z_{u}$, there are at least
$h(t)-\left\lfloor\frac{h(t)}{h(u)}\right\rfloor$ colors remaining for $t$,
all of which ensure that $u$ guesses incorrectly. Finally, there are
$\left\lceil
h(w)\left(1-\left\lceil\frac{h(v)}{h(u)}\right\rceil^{-1}\right)\right\rceil$
colors for $w$ that ensure $v$ guesses incorrectly. Hence, after the vertices
of $C$ commit to a guessing strategy, we can commit to coloring the vertices
of $C\backslash\\{u,v\\}$ using $h^{\prime}(x)$ colors for each $x\in
C\backslash\\{u,v\\}$ in a way that guarantees $u$ and $v$ guess incorrectly.
Hence, if $\langle C,h\rangle$ is winning then the corresponding winning
strategy must give a winning strategy on $\langle
C\backslash\\{u,v\\},h^{\prime}\rangle$. ∎

Our next lemma handles a step present in a few cases in Theorem 1.1.
Equivalent results have been shown in other works, but we reprove the result
here in terms that are useful for our arguments. The lemma can be thought of
as a “vertex deletion” lemma as it allows us to delete a vertex with hatness
$5$ lying on a path without changing the winningness of the game.

######  Lemma 2.4.

Let $P$ be a path and let $h$ be a hatness function with $h(v)=5$ for some
$v\in P$. The game $\langle P,h\rangle$ is winning if and only if there is a
connected proper subgraph of $P$ which is winning.

###### Proof.

Let $P$ be a path and let $h$ be a hatness function with $h(v)=5$ for some
$v\in P$. If $P$ contains a winning connected proper subgraph then $P$ is
winning. Suppose $P$ does not contain a winning connected proper subgraph and
let $L,R$ be the resulting disjoint paths when $v$ is deleted from $P$. Then
$L$ and $R$ are connected proper subgraphs of $P$, so they are losing with
their given hatness functions. The three vertex path with hatnesses $(2,5,2)$
is also losing. By Theorem 2.2, we can glue together this path with $L$ on one
endpoint and $R$ on the other endpoint to get a new losing path where the
hatnesses are inherited from the endpoints of $L$ and $R$. Note that this is
exactly the game $\langle P,h\rangle$, so $P$ is losing. ∎

Now, we are ready to prove Theorem 1.1 when Conditions 1, 2, 3, and 4b all
fail. In particular, when Condition 4b fails, there is a vertex with hatness
at least $5$.

###### Proof.

Let $C$ be a cycle of length at least $4$ with at least one vertex of hatness
$5$ and no winning proper subgraph. Observe that if there are no vertices with
hatness $2$, the game is losing by [15, Theorem 1]. Hence, we may assume there
is at least one vertex with hatness $2$. We may also assume without loss of
generality that every vertex has hatness $2$, $3$, or $5$ since decreasing the
hatness of a vertex does not make a winning game losing. Finally, on any path
between two vertices of hatness $2$, we assume there is at most one vertex
with hatness $5$ for the same reason.

If there is a vertex with hatness $2$ adjacent to a vertex of hatness $5$ then
there is one of the following sequences of hatnesses: $(2,5,2)$, $(5,2,3,5)$,
$(5,2,3,3)$. Otherwise, there is no vertex with hatness $2$ next to a vertex
of hatness $5$, and we must have a sequence $(3,2,3,3)$ or $(3,2,3,5)$ in the
cycle. It is easy to see that these cover all possible cases, with the
exception of the exclusion of the case $(5,2,5,3)$. This can be attributed to
the fact that the presence of two vertices of hatness $5$ indicates the
presence of a vertex of hatness $2$ somewhere else in the cycle, with at most
one vertex of hatness $5$ as a neighbor. Now, we handle each case.

For the sequence $(2,5,2)$, we can choose a color for the vertex of hatness
$5$ that is never guessed, leaving a path that induces proper subgame of
$\langle C,h\rangle$. Hence, this path must be losing, and the result holds.

For the sequence $(5,2,3,5)$, we apply Lemma 2.3 with

$h(t)=5,\,h(u)=2,\,h(v)=3,\,h(w)=5$

to get a path with endpoints $t$ and $w$ with $h^{\prime}(t)=3$ and
$h^{\prime}(w)=3$. Observe that $t$ and $w$ have degree one in the new graph,
and so they are deletable by Theorem 2.2. This leaves a proper subgame that
must must be losing. Hence, $\langle C,h\rangle$ is losing.

For the sequence $(5,2,3,3)$, we again apply Lemma 2.3 with

$h(t)=5,\,h(u)=2,\,h(v)=3,\,h(w)=3$

to get a path with endpoints $t$ and $w$ with $h^{\prime}(t)=3$ and
$h^{\prime}(w)=2$. Observe that $t$ has degree one in the new graph, and so it
is deletable. Suppose toward a contradiction that the new path is winning.
Then, by Theorem 2.1 we can glue one end of a path $(2,3,2)$ which is winning
to $w$ and create a new graph that is still winning but where $w$ has hatness
$4$. Then, we can decrease the hatness of $w$ to $3$, and the graph is still
winning. However, this new graph is a proper subgraph of the original cycle
$C$, which yields a contradiction. Hence, $\langle C,h\rangle$ is losing.

For the sequence $(3,2,3,3)$, we apply Lemma 2.3 with

$h(t)=3,\,h(u)=2,\,h(v)=3,\,h(w)=3$

to get $h^{\prime}(t)=2$ and $h^{\prime}(w)=2$. Suppose toward a contradiction
that the resulting path is winning. Then we can glue a copy of the winning
path with hatnesses $(2,2)$ to $t$ and $(2,3,2)$ to $w$, which gives a new
winning path where $t$ and $w$ both have hatness $4$ by Theorem 2.1.
Decreasing the hatnesses for $t$ and $w$ to $3$ preserves the winningness of
the path. Note that this path contains a vertex of hatness $5$. By Lemma 2.4,
there must be a winning connected proper subgraph. However, any connected
proper subgraph of this path cannot contain both endpoints and is thus also a
proper subgraph of the original cycle, yielding a contradiction. Hence,
$\langle C,h\rangle$ is losing.

For the sequence $(3,2,3,5)$, we apply Lemma 2.3 with

$h(t)=3,\,h(u)=2,\,h(v)=3,\,h(w)=5$

to get $h^{\prime}(t)=2$ and $h^{\prime}(w)=3$. Since $w$ has hatness $3$ in
the new graph, $w$ is deletable. After deleting $w$, suppose toward a
contradiction that the graph is winning. Then we can glue an edge with
hatnesses $(2,2)$ to $t$, creating a winning graph where $t$ has hatness $4$.
We can then decrease the hatness of $t$ to $3$. This graph is now a proper
subgame of $\langle C,h\rangle$, which must be losing, a contradiction. Hence,
$\langle C,h\rangle$ is losing. ∎

##  3 Cactus Graphs

We start by proving the general upper bound that every cactus graph has hat
guessing number at most $4$, which proves Theorem 1.2, Statement 1. Then, we
show how the same argument can be modified to prove Statement 2. Finally, we
observe that Statement 3 follows immediately from [10].

######  Definition 3.1.

Let $G$ be a cactus graph with a cycle $C$. We say that $C$ is a _leaf cycle_
if deleting the edges of $C$ leaves at most $1$ connected component with a
cycle (so all other connected components are acyclic).

Figure 5 depicts the leaf cycles of a cactus graph in bold.

Figure 5: A cactus graph with the leaf cycles in bold.

###### Proof of Theorem 1.2.

First, we prove the general upper bound of $HG(G)\leq 4$ for all cactus graphs
$G$ by induction on the number of vertices. The base case is trivial. Suppose
that the result holds for all cactus graphs on $n\geq 1$ vertices, and let $G$
be a cactus graph with $n+1$ vertices. We show that $HG(G)<5$. If $G$ has a
vertex $v$ with degree $1$ then [1, Theorem 1.8] shows that
$HG(G)=HG(G\backslash\\{v\\})$, and the result holds by induction. Thus, we
may assume that $G$ has no vertices of degree $1$. If $G$ has no cycles then
$G$ is a tree, and $HG(G)=2$. Hence, we may also assume $G$ has at least one
cycle. In particular, this implies $G$ has a non-empty leaf cycle $C$. We can
decompose $G$ as $G=C+_{v,v}G^{\prime}$ at a vertex $v\in C\cap G^{\prime}$,
where $G^{\prime}$ is a cactus graph on fewer than $n+1$ vertices. By the
induction hypothesis, $HG(G^{\prime})<5$, so the game $\langle
G^{\prime},\star 5\rangle$ is losing. Theorem 1.1 shows that $\langle
C,h\rangle$ is losing where $h(v)=2$ and $h(w)=5$ for all $w\neq v$. Then
$\langle G,\star 5\rangle$ is losing by Theorem 2.2, so $HG(G)<5$ and the
result holds.

The same argument can easily be modified to prove the upper bound for
Statement 2, by assuming $HG(G)<4$ in the induction hypothesis for graphs
satisfying Statement 2. If the graph has exactly one cycle, then it is a
pseudotree and the result holds by [10]. Otherwise, $G$ has at least two
cycles and thus, at least two leaf cycles. Statement 2 ensures at most one
triangle, so we can choose a non-triangle leaf cycle $C$ in $G$ and write
$G=C+_{v}G^{\prime}$, where $G^{\prime}$ is a cactus graph on fewer than $n+1$
vertices. Then $\langle C,h\rangle$ is losing where $h(v)=2$ and $h(w)=4$ for
all $w\neq v$, and $\langle G^{\prime},\star 4\rangle$ is losing by the
induction hypothesis, so $\langle G,\star 4\rangle$ is losing by Theorem 2.2.

As Statement 3 follows from [10], the proof of Theorem 1.2 is complete. ∎

##  4 Open Problems

Theorem 1.1 (along with some lemmas about leaf deletion) can extend the result
for pseudotrees [10] to arbitrary hatness functions [8]. Naturally, we would
like to do so for graphs with more than one cycle. The treelike structure of
cactus graphs, which this paper leveraged, could help make them the next step.
One challenge that must be overcome is that the hatness for any individual
vertex of a winning cactus graph may be arbitrarily large. For example, we can
take $n$ copies of the triangle with hatnesses $(2,4,4)$ and glue them
together at the vertices of hatness $2$. The resulting graph is a winning
cactus with a vertex of hatness $2^{n}$. Even worse, we can glue multiple
copies of this graph together to get arbitrarily many vertices with hatness
$2^{n}$.

Instead, we could turn to _Theta graphs_ , consisting of three internally
vertex disjoint paths that share common endpoints. Thetas are the “simplest”
non-cactus graphs, and yet we lack even an analogue to Theorem 1.2. Every
theta has maximum degree $3$, so the hat guessing number of a theta is
trivially at most $8$ by [5]. Careful analysis using a generalized version of
Lemma 2.3 shows that thetas have hat guessing number at most $4$. We postpone
a full characterization of thetas for both constant and general hatness
functions to a future work.

Acknowledgment: We thank Noga Alon for connecting the authors and Stephen
Melczer for helpful suggestions.

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