On the Chromatic Polynomial and Counting DP-Colorings

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. The list color function of graph $G$, denoted $P_{\ell}(G,m)$, is a list analogue of the chromatic polynomial that has been studied since 1992, primarily through comparisons with the corresponding chromatic polynomial. DP-coloring (also called correspondence coloring) is a generalization of list coloring recently introduced by Dvo\v{r}\'{a}k and Postle. In this paper, we introduce a DP-coloring analogue of the chromatic polynomial called the DP color function, denoted $P_{DP}(G,m)$, and ask several fundamental open questions about it, making progress on some of them. Motivated by known results related to the list color function, we show that while the DP color function behaves similar to the list color function for some graphs, there are also some surprising differences. For example, Wang, Qian, and Yan recently showed that if $G$ is a connected graph with $l$ edges, then $P_{\ell}(G,m)=P(G,m)$ whenever $m > \frac{l-1}{\ln(1+ \sqrt{2})}$, but we will show that for any $g \geq 3$ there exists a graph, $G$, with girth $g$ such that $P_{DP}(G,m) < P(G,m)$ when $m$ is sufficiently large. We also study the asymptotics of $P(G,m) - P_{DP}(G,m)$ for a fixed graph $G$. We develop techniques to compute $P_{DP}(G,m)$ exactly and apply them to certain classes of graphs such as chordal graphs, unicyclic graphs, and cycles with a chord. Finally, we make progress towards showing that for any graph $G$, there is a $p$ such that $P_{DP}(G \vee K_p, m) = P(G \vee K_p , m)$ for large enough $m$.Comment: 23 page

The List Distinguishing Number Equals the Distinguishing Number for Interval Graphs

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$ such that $G$ has a distinguishing coloring where each vertex is assigned a color from $\{1,\ldots,k\}$. A \textit{list assignment} to $G$ is an assignment $L=\{L(v)\}_{v\in V(G)}$ of lists of colors to the vertices of $G$. A \textit{distinguishing $L$-coloring} of $G$ is a distinguishing coloring of $G$ where the color of each vertex $v$ comes from $L(v)$. The {\it list distinguishing number} of $G$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)|=k$ for all $v\in V(G)$ yields a distinguishing $L$-coloring of $G$. We prove that if $G$ is an interval graph, then its distinguishing number and list distinguishing number are equal.Comment: 11 page

Spatial Mixing of Coloring Random Graphs

We study the strong spatial mixing (decay of correlation) property of proper $q$-colorings of random graph $G(n, d/n)$ with a fixed $d$. The strong spatial mixing of coloring and related models have been extensively studied on graphs with bounded maximum degree. However, for typical classes of graphs with bounded average degree, such as $G(n, d/n)$, an easy counterexample shows that colorings do not exhibit strong spatial mixing with high probability. Nevertheless, we show that for $q\ge\alpha d+\beta$ with $\alpha>2$ and sufficiently large $\beta=O(1)$, with high probability proper $q$-colorings of random graph $G(n, d/n)$ exhibit strong spatial mixing with respect to an arbitrarily fixed vertex. This is the first strong spatial mixing result for colorings of graphs with unbounded maximum degree. Our analysis of strong spatial mixing establishes a block-wise correlation decay instead of the standard point-wise decay, which may be of interest by itself, especially for graphs with unbounded degree

List Coloring a Cartesian Product with a Complete Bipartite Factor

We study the list chromatic number of the Cartesian product of any graph $G$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $\chi_\ell(G \square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $\chi_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since $\chi_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells us the values of $b$ for which $\chi_\ell(K_{a,b})$ is as large as possible and far from $\chi(K_{a,b})=2$. In this paper we seek to understand when $\chi_\ell(G \square K_{a,b})$ is far from $\chi(G \square K_{a,b}) = \max \{\chi(G), 2 \}$. It is easy to show $\chi_\ell(G \square K_{a,b}) \leq \chi_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Kr\'al, and Miskuf showed that this bound is attainable if $b$ is sufficiently large; specifically, $\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$ whenever $b \geq (\chi_\ell(G) + a - 1)^{a|V(G)|}$. Given any graph $G$ and $a \in \mathbb{N}$, we wish to determine the smallest $b$ such that $\chi_\ell(G \square K_{a,b}) = \chi_\ell (G) + a$. In this paper we show that the list color function, a list analogue of the chromatic polynomial, provides the right concept and tool for making progress on this problem. Using the list color function, we prove a general improvement on Borowiecki et al.'s 2006 result, and we compute the smallest such $b$ exactly for some large families of chromatic-choosable graphs.Comment: 12 page

Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page

Brooks's theorem for measurable colorings

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $\mu$-measurable $d$-coloring with respect to any Borel probability measure $\mu$ on $X$, and a Baire measurable $d$-coloring with respect to any compatible Polish topology on $X$. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID $d$-colorings of Cayley graphs of degree $d$, except in two exceptional cases.Comment: Minor correction

On the Connectedness of Clash-free Timetables

We investigate the connectedness of clash-free timetables with respect to the Kempe-exchange operation. This investigation is related to the connectedness of the search space of timetabling problem instances, which is a desirable property, for example for two-step algorithms using the Kempe-exchange during the optimization step. The theoretical framework for our investigations is based on the study of reconfiguration graphs, which model the search space of timetabling problems. We contribute to this framework by including timeslot availability requirements in the analysis and we derive improved conditions for the connectedness of clash-free timetables in this setting. We apply the theoretical insights to establish the connectedness of clash-free timetables for a number of benchmark instances

A reverse Sidorenko inequality

Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality $$\hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)},$$ where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has $$\hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2}$$ for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of $G$ is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to $H = K_q$, we show that the triangle-free hypothesis on $G$ may be dropped; this is also valid if some of the vertices of $K_q$ are looped. A corollary is that among $d$-regular graphs, $G = K_{d,d}$ maximizes the quantity $c_q(G)^{1/|V(G)|}$ for every $q$ and $d$, where $c_q(G)$ counts proper $q$-colorings of $G$. Finally, we show that if the edge-weight matrix of $H$ is positive semidefinite, then $$\hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}.$$ This implies that among $d$-regular graphs, $G = K_{d+1}$ maximizes $\hom(G, H)^{1/|V(G)|}$. For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikv\'ari-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn.Comment: 30 page

3 List Coloring Graphs of Girth at least Five on Surfaces

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this result to prove Havel's conjecture that a planar graph whose triangles are pairwise far enough apart is 3-colorable. As for list-coloring, Dvorak proved that a planar graph whose cycles of size at most four are pairwise far enough part is 3-choosable. In this article, we generalize these results. First we prove a linear isoperimetric bound for 3-list-coloring graphs of girth at least five. Many new results then follow from the theory of hyperbolic families of graphs developed by Postle and Thomas. In particular, it follows that there are only finitely many 4-list-critical graphs of girth at least five on any fixed surface, and that in fact the number of vertices of a 4-list-critical graph is linear in its genus. This provides independent proofs of the above results while generalizing Dvorak's result to graphs on surfaces that have large edge-width and yields a similar result showing that a graph of girth at least five with crossings pairwise far apart is 3-choosable. Finally, we generalize to surfaces Thomassen's result that every planar graph of girth at least five has exponentially many distinct 3-list-colorings. Specifically, we show that every graph of girth at least five that has a 3-list-coloring has $2^{\Omega(n)-O(g)}$ distinct 3-list-colorings.Comment: 33 page

Strong spatial mixing for list coloring of graphs

The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for regular trees when $q \geq \alpha^{*} \Delta + 1$ where $q$ the number of colors, $\Delta$ is the degree and $\alpha^* = 1.763..$ is the unique solution to $xe^{-1/x} = 1$. It has also been established for bounded degree lattice graphs whenever $q \geq \alpha^* \Delta - \beta$ for some constant $\beta$, where $\Delta$ is the maximum vertex degree of the graph. The latter uses a technique based on recursively constructed coupling of Markov chains whereas the former is based on establishing decay of correlations on the tree. We establish strong spatial mixing of list colorings on arbitrary bounded degree triangle-free graphs whenever the size of the list of each vertex $v$ is at least $\alpha \Delta(v) + \beta$ where $\Delta(v)$ is the degree of vertex $v$ and $\alpha > \alpha ^*$ and $\beta$ is a constant that only depends on $\alpha$. We do this by proving the decay of correlations via recursive contraction of the distance between the marginals measured with respect to a suitably chosen error function