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Jeffrey A. Mudrock

Publications and source records attributed to Jeffrey A. Mudrock.

At least 37 records · Page 2Linked to original sources

arXiv2024

On Polynomial Representations of Dual DP Color Functions

DP-coloring (also called correspondence coloring) is a generalization of list coloring that was introduced by Dvořák and Postle in 2015. The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted $P(G,m)$, it equals the number of proper $m$-colorings of graph $G$. Counting function analogues of chromatic polynomials have been introduced for list colorings: $P_{\ell}$, list color functions (1990); DP colorings: $P_{DP}$, DP color functions (2019), and $P^*_{DP}$, dual DP color functions (2021). For any graph $G$ and $m \in \mathbb{N}$, $P_{DP}(G, m) \leq P_\ell(G,m) \leq P(G,m) \leq P_{DP}^*(G,m)$. In 2022 (improving on older results) Dong and Zhang showed that for any graph $G$, $P_{\ell}(G,m)=P(G,m)$ whenever $m \geq |E(G)|-1$. Consequently, the list color function of a graph is a polynomial for sufficiently large $m$. One of the most important and longstanding open questions on DP color functions asks: for every graph $G$ is there an $N \in \mathbb{N}$ and a polynomial $p(m)$ such that $P_{DP}(G,m) = p(m)$ whenever $m \geq N$? We show that the answer to the analogue of this question for dual DP color functions is no. Our proof reveals a connection between a dual DP color function and the balanced chromatic polynomial of a signed graph introduced by Zaslavsky in 1982.

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arXiv2024

A Note on Fractional DP-Coloring of Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that unlike the fractional list chromatic number, the fractional DP-chromatic number of a graph $G$, denoted $χ_{_{DP}}^*(G)$, can be arbitrarily larger than $χ^*(G)$, the graph's fractional chromatic number. We generalize a result of Alon, Tuza, and Voigt (1997) on the fractional list chromatic number of odd cycles, and, in the process, show that for each $k \in \mathbb{N}$, $χ_{_{DP}}^*(C_{2k+1}) = χ^*(C_{2k+1})$. We also show that for any $n \geq 2$ and $m \in \mathbb{N}$, if $p^*$ is the solution in $(0,1)$ to $p=(1-p)^n$ then $χ_{_{DP}}^*(K_{n,m})\leq1/p^*$, and we prove a generalization of this result for multipartite graphs. Finally, we determine a lower bound on $χ_{_{DP}}^*(K_{2,m})$ for any $m \geq 3$.

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arXiv2024

Flexible list colorings: Maximizing the number of requests satisfied

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose $0 \leq ε\leq 1$, $G$ is a graph, $L$ is a list assignment for $G$, and $r$ is a function with non-empty domain $D\subseteq V(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of $L$). The triple $(G,L,r)$ is $ε$-satisfiable if there exists a proper $L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $ε|D|$ vertices in $D$. We say $G$ is $(k, ε)$-flexible if $(G,L',r')$ is $ε$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is a request of $L'$. It was shown by Dvořák et al. that if $d+1$ is prime, $G$ is a $d$-degenerate graph, and $r$ is a request for $G$ with domain of size $1$, then $(G,L,r)$ is $1$-satisfiable whenever $L$ is a $(d+1)$-assignment. In this paper, we extend this result to all $d$ for bipartite $d$-degenerate graphs. The literature on flexible list coloring tends to focus on showing that for a fixed graph $G$ and $k \in \mathbb{N}$ there exists an $ε> 0$ such that $G$ is $(k, ε)$-flexible, but it is natural to try to find the largest possible $ε$ for which $G$ is $(k,ε)$-flexible. In this vein, we improve a result of Dvořák et al., by showing $d$-degenerate graphs are $(d+2, 1/2^{d+1})$-flexible. In pursuit of the largest $ε$ for which a graph is $(k,ε)$-flexible, we observe that a graph $G$ is not $(k, ε)$-flexible for any $k$ if and only if $ε> 1/ ρ(G)$, where $ρ(G)$ is the Hall ratio of $G$, and we initiate the study of the list flexibility number of a graph $G$, which is the smallest $k$ such that $G$ is $(k,1/ ρ(G))$-flexible. We study relationships and connections between the list flexibility number, list chromatic number, list packing number, and degeneracy of a graph.

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arXiv2023

The DP Color Function of Clique-Gluings of Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. Formulas for chromatic polynomials of clique-gluings of graphs, a fundamental graph operation, are well-known, but the effect of such gluings on the DP color function is not well understood. In this paper we study the DP color function of $K_p$-gluings of graphs. Recently, Becker et. al. asked whether $P_{DP}(G,m) \leq (\prod_{i=1}^n P_{DP}(G_i,m))/\left( \prod_{i=0}^{p-1} (m-i) \right)^{n-1}$ whenever $m \geq p$, where the expression on the right is the DP-coloring analogue of the corresponding chromatic polynomial formula for a $K_p$-gluing, $G$, of $G_1, \ldots, G_n$. Becker et. al. showed this inequality holds when $p=1$. In this paper we show this inequality holds for edge-gluings ($p=2$). On the other hand, we show it does not hold for triangle-gluings ($p=3$), which also answers a question of Dong and Yang (2021). Finally, we show a relaxed version, based on a class of $m$-fold covers that we conjecture would yield the fewest DP-colorings for a given graph, of the inequality holds when $p \geq 3$.

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arXiv2023

On Polynomial Representations of the DP Color Function: Theta Graphs and Their Generalizations

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial $P(G,m)$, the DP color function of a graph $G$, denoted $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. It is known that, unlike the list color function $P_{\ell}(G,m)$, 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. Thus, two fundamental open questions regarding the DP color function are: (i) for which $G$ does there exist an $N \in \mathbb{N}$ such that $P_{DP}(G,m) = P(G,m)$ whenever $m \geq N$, (ii) Given a graph $G$ does there always exist an $N \in \mathbb{N}$ and a polynomial $p(m)$ such that $P_{DP}(G,m) = p(m)$ whenever $m \geq N$? In this paper we give exact formulas for the DP color function of a Theta graph based on the parity of its path lengths. This gives an explicit answer, including the formulas for the polynomials that are not the chromatic polynomial, to both the questions above for Theta graphs. We extend this result to Generalized Theta graphs by characterizing the exact parity condition that ensures the DP color function eventually equals the chromatic polynomial. To answer the second question for Generalized Theta graphs, we confirm it for the larger class of graphs with a feedback vertex set of size one.

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arXiv2023

On the List Color Function Threshold

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 the early 1990s, primarily through comparisons with the corresponding chromatic polynomial. It is known that for any graph $G$ there is a $k \in \mathbb{N}$ such that $P_\ell(G,m) = P(G,m)$ whenever $m \geq k$. The list color function threshold of $G$, denoted $τ(G)$, is the smallest $k \geq χ(G)$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. In 2009, Thomassen asked whether there is a universal constant $α$ such that for any graph $G$, $τ(G) \leq χ_{\ell}(G) + α$, where $χ_{\ell}(G)$ is the list chromatic number of $G$. We show that the answer to this question is no by proving that there exists a constant $C$ such that $τ(K_{2,l}) - χ_{\ell}(K_{2,l}) \ge C\sqrt{l}$ for $l \ge 16$.

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arXiv2023

An Algebraic Approach for Counting DP-3-colorings of Sparse Graphs

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$, and the list color function, $P_{\ell}(G,m)$, the DP color function of $G$, denoted by $P_{DP}(G,m)$, counts the minimum number of DP-colorings over all possible $m$-fold covers. It follows that $P_{DP}(G,m) \le P_{\ell}(G,m) \le P(G,m)$. A function $f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a \geq χ(G)$ implies that $f(G,m) = P(G,m)$ for all $m \geq a$. It is known that the DP color function is not chromatic-adherent, but there are only two known graphs that demonstrate this. Suppose $G$ is an $n$-vertex graph and $\mathcal{H}$ is a 3-fold cover of $G$, in this paper we associate with $\mathcal{H}$ a polynomial $f_{G, \mathcal{H}} \in \mathbb{F}_3[x_1, \ldots, x_n]$ so that the number of non-zeros of $f_{G, \mathcal{H}}$ equals the number of $\mathcal{H}$-colorings of $G$. We then use a well-known result of Alon and Füredi on the number of non-zeros of a polynomial to establish a non-trivial lower bound on $P_{DP}(G,3)$ when $2n > |E(G)|$. An easy consequence of this is that $P_{DP}(G, 3) \geq 3^{n/6}$ for every $n$-vertex planar graph $G$ of girth at least 5, improving the previously known bounds on both $P_{DP}(G, 3)$ and $P_{\ell}(G, 3)$. Finally, we use this bound to show that there are infinitely many graphs that demonstrate the non-chromatic-adherence of the DP color function.

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arXiv2022

Bounding the List Color Function Threshold from Above

The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$ for each $m \in \mathbb{N}$. In 1990, Kostochka and Sidorenko introduced the list color function of graph $G$, denoted $P_{\ell}(G,m)$, which is a list analogue of the chromatic polynomial. The list color function threshold of $G$, denoted $τ(G)$, is the smallest $k \geq χ(G)$ such that $P_{\ell}(G,m) = P(G,m)$ whenever $m \geq k$. It is known that for every graph $G$, $τ(G)$ is finite, and in fact, $τ(G) \leq (|E(G)|-1)/\ln(1+ \sqrt{2}) + 1$. It is also known that when $G$ is a cycle or chordal graph, $G$ is enumeratively chromatic-choosable which means $τ(G) = χ(G)$. A recent paper of Kaul et al. suggests that understanding the list color function threshold of complete bipartite graphs is essential to the study of the extremal behavior of $τ$. In this paper we show that for any $n \geq 2$, $τ(K_{2,n}) \leq \lceil (n+2.05)/1.24 \rceil$ which gives an improvement on the general upper bound for $τ(G)$ when $G = K_{2,n}$. We also develop additional tools that allow us to show that $τ(K_{2,3}) = χ(K_{2,3})$ and $τ(K_{2,4}) = τ(K_{2,5}) = 3$.

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arXiv2022

A Short Proof that the List Packing Number of any Graph is Well Defined

List packing is a notion that was introduced in 2021 (by Cambie et al.). The list packing number of a graph $G$, denoted $χ_{\ell}^*(G)$, is the least $k$ such that for any list assignment $L$ that assigns $k$ colors to each vertex of $G$, there is a set of $k$ proper $L$-colorings of $G$, $\{f_1, \ldots, f_k \}$, with the property $f_i(v) \neq f_j(v)$ whenever $1 \leq i < j \leq k$ and $v \in V(G)$. We present a short proof that for any graph $G$, $χ_{\ell}^*(G) \leq |V(G)|$. Interestingly, our proof makes use of Galvin's celebrated result that the list chromatic number of the line graph of any bipartite multigraph equals its chromatic number.

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arXiv2022

DP-Coloring Cartesian Products of Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. Motivated by results related to list coloring Cartesian products of graphs, we initiate the study of the DP-chromatic number, $χ_{DP}$, of the same. We show that $χ_{DP}(G \square H) \leq \text{min}\{χ_{DP}(G) + \text{col}(H), χ_{DP}(H) + \text{col}(G) \} - 1$ where $\text{col}(H)$ is the coloring number of the graph $H$. We focus on building tools for lower bound arguments for $χ_{DP}(G \square H)$ and use them to show the sharpness of the bound above and its various forms. Our results illustrate that the DP color function of $G$, the DP analogue of the chromatic polynomial, is essential in the study of the DP-chromatic number of the Cartesian product of graphs, including the following question that extends the sharpness problem above and the classical result on gap between list chromatic number and chromatic number: given any graph $G$ and $k \in \mathbb{N}$, what is the smallest $t$ for which $χ_{DP}(G \square K_{k,t})= χ_{DP}(G) + k$?

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arXiv2022

The DP Color Function of Joins and Vertex-Gluings of Graphs

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arXiv2021

On the Equitable Choosability of the Disjoint Union of Stars

Equitable $k$-choosability is a list analogue of equitable $k$-coloring that was introduced by Kostochka, Pelsmajer, and West in 2003. It is known that if vertex disjoint graphs $G_1$ and $G_2$ are equitably $k$-choosable, the disjoint union of $G_1$ and $G_2$ may not be equitably $k$-choosable. Given any $m \in \mathbb{N}$ the values of $k$ for which $K_{1,m}$ is equitably $k$-choosable are known. Also, a complete characterization of equitably $2$-choosable graphs is not known. With these facts in mind, we study the equitable choosability of $\sum_{i=1}^n K_{1,m_i}$, the disjoint union of $n$ stars. We show that determining whether $\sum_{i=1}^n K_{1,m_i}$ is equitably choosable is NP-complete when the same list of two colors is assigned to every vertex. We completely determine when the disjoint union of two stars (or $n \geq 2$ identical stars) is equitably 2-choosable, and we present results on the equitable $k$-choosability of the disjoint union of two stars for arbitrary $k$.

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arXiv2021

Non-chromatic-adherence of the DP Color Function via Generalized Theta Graphs

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvořák and Postle in 2015. The chromatic polynomial of a graph is an extensively studied notion in combinatorics since its introduction by Birkhoff in 1912; denoted $P(G,m)$, it equals the number of proper $m$-colorings of graph $G$. Counting function analogues of the chromatic polynomial have been introduced and studied for list colorings: $P_{\ell}$, the list color function (1990); DP colorings: $P_{DP}$, the DP color function (2019), and $P^*_{DP}$, the dual DP color function (2021). For any graph $G$ and $m \in \mathbb{N}$, $P_{DP}(G, m) \leq P_\ell(G,m) \leq P(G,m) \leq P_{DP}^*(G,m)$. A function $f$ is chromatic-adherent if for every graph $G$, $f(G,a) = P(G,a)$ for some $a \geq χ(G)$ implies that $f(G,m) = P(G,m)$ for all $m \geq a$. It is not known if the list color function and the DP color function are chromatic-adherent. We show that the DP color function is not chromatic-adherent by studying the DP color function of Generalized Theta graphs. The tools we develop along with the Rearrangement Inequality give a new method for determining the DP color function of all Theta graphs and the dual DP color function of all Generalized Theta graphs.

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arXiv2021

A Deletion-Contraction Relation for the DP Color Function

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph $G$, denoted $P(G,m)$, is equal to the number of proper $m$-colorings of $G$. A well-known tool for computing the chromatic polynomial of graph $G$ is the deletion-contraction formula which relates $P(G,m)$ to the chromatic polynomials of two smaller graphs. The DP color function of a graph $G$, denoted $P_{DP}(G,m)$, is a DP-coloring analogue of the chromatic polynomial, and $P_{DP}(G,m)$ is the minimum number of DP-colorings of $G$ over all possible $m$-fold covers. In this paper we present a deletion-contraction relation for the DP color function. To make this possible, we extend the definition of the DP color function to multigraphs. We also introduce the dual DP color function of a graph $G$, denoted $P^*_{DP}(G,m)$, which counts the maximum number of DP-colorings of $G$ over certain $m$-fold covers. We show how the dual DP color function along with our deletion-contraction relation yields a new general lower bound on the DP color function of a graph.

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arXiv2021

On Equitable List Arboricity of Graphs

Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring the subgraph induced by each color class to be acyclic (instead of edgeless) in addition to the usual upper bound on the size of each color class. Graph $G$ is equitably $k$-list arborable if an equitable, arborable list coloring of $G$ exists for every list assignment for $G$ that associates with each vertex in $G$ a list of $k$ available colors. Zhang conjectured that any graph $G$ is equitably $k$-list arborable for each $k$ satisfying $k \geq \lceil (1+Δ(G))/2 \rceil$. We verify this conjecture for powers of cycles by applying a new lemma which is a general tool for extending partial equitable, arborable list colorings. We also propose a stronger version of Zhang's Conjecture for certain connected graphs: any connected graph $G$ is equitably $k$-list arborable for each $k$ satisfying $k \geq \lceil Δ(G)/2 \rceil$ provided $G$ is neither a cycle nor a complete graph of odd order. We verify this stronger version of Zhang's Conjecture for powers of paths, 2-degenerate graphs, and certain other graphs. We also show that if $G$ is equitably $k$-list arborable it does not necessarily follow that $G$ is equitably $(k+1)$-list arborable which addresses a question of Drgas-Burchardt, Furmanczyk, and Sidorowicz (2018).

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arXiv2021

Partial DP-Coloring

In 1980, Albertson and Berman introduced partial coloring. In 2000, Albertson, Grossman, and Haas introduced partial list coloring. Here, we initiate the study of partial coloring for an insightful generalization of list coloring introduced in 2015 by Dvořák and Postle, DP-coloring (or correspondence coloring). We consider the DP-coloring analogue of the Partial List Coloring Conjecture, which generalizes a natural bound for partial coloring. We show that while this partial DP-coloring conjecture does not hold, several results on partial list coloring can be extended to partial DP-coloring. We also study partial DP-coloring of the join of a graph with a complete graph, and we present several interesting open questions.

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arXiv2020

Answers to Two Questions on the DP Color Function

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic polynomial of graph $G$ is denoted $P(G,m)$, and it is equal to the number of proper $m$-colorings of $G$. In 2019, Kaul and Mudrock introduced an analogue of the chromatic polynomial for DP-coloring; specifically, the DP color function of graph $G$ is denoted $P_{DP}(G,m)$. Two fundamental questions posed by Kaul and Mudrock are: (1) For any graph $G$ with $n$ vertices, is it the case that $P(G,m)-P_{DP}(G,m) = O(m^{n-3})$ as $m \rightarrow \infty$? and (2) For every graph $G$, does there exist $p,N \in \mathbb{N}$ such that $P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)$ whenever $m \geq N$? We show that the answer to both these questions is yes. In fact, we show the answer to (2) is yes even if we require $p=1$.

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arXiv2020

Combinatorial Nullstellensatz and DP-coloring of Graphs

We initiate the study of applying the Combinatorial Nullstellensatz to the DP-coloring of graphs even though, as is well-known, the Alon-Tarsi theorem does not apply to DP-coloring. We define the notion of good covers of prime order which allows us to apply the Combinatorial Nullstellensatz to DP-coloring. We apply these tools to DP-coloring of the cones of certain bipartite graphs and uniquely 3-colorable graphs. We also extend a result of Akbari, Mirrokni, and Sadjad (2006) on unique list colorability to the context of DP-coloring. We establish a sufficient algebraic condition for a graph $G$ to satisfy $χ_{DP}(G) \leq 3$, and we completely determine the DP-chromatic number of squares of all cycles.

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