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Hemanshu Kaul

Publications and source records attributed to Hemanshu Kaul.

At least 19 recordsLinked to original sources
arXiv2026

Counting List Colorings of Unlabeled Graphs

The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph $\mathcal{G}$, $P(\mathcal{G}, k)$. Determining $P(\mathcal{G}, k)$ amounts to counting colorings under the action of automorphisms of $\mathcal{G}$. In this paper, we consider the problem of counting list colorings of unlabeled graphs. We extend Hanlon's definition to the list context and define the unlabeled list color function, $P_\ell(\mathcal{G}, k)$, of an unlabeled graph $\mathcal{G}$. In this context, we pursue a fundamental question whose analogues have driven much of the research on counting list colorings and its generalizations: For a given unlabeled graph $\mathcal{G}$, does $P_\ell(\mathcal{G}, k) = P(\mathcal{G}, k)$ when $k$ is large enough? We show the answer to this question is yes for almost all graphs, in particular, for a large class of unlabeled graphs that includes point-determining graphs (also known as twin-free graphs, irreducible graphs, and mating graphs).

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arXiv2026

The DP Color Function of Bipartite 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 $P(G,q)$, the chromatic polynomial of a graph $G$, the DP color function of $G$, denoted by $P_{DP}(G,q)$, counts the minimum number of DP-colorings over all $q$-fold covers of $G$. It follows that $P_{DP}(G,q) \leq P(G,q)$. It is known that there are graphs for which $P_{DP}(G,q) < P(G,q)$ for all sufficiently large $q$; in fact, all bipartite graphs containing a cycle have this property. A fundamental open question about DP color functions asks whether, for every graph $G$, there exist $N \in \mathbb{N}$ and a polynomial $p$ such that $P_{DP}(G,q) = p(q)$ whenever $q \geq N$. In this paper we answer this question affirmatively for all bipartite graphs. Specifically, if $G$ is an $n$-vertex bipartite graph with $c$ components, then $P_{DP}(G,q) = (-1)^{n-c}q^c \;T_G(1-q,1)$ for all sufficiently large $q$, where $T_G(x,y)$ is the Tutte polynomial of $G$. The ideas we develop also yield an asymptotic formula for $P(G,q)-P_{DP}(G,q)$ whenever the girth of $G$ is even.

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arXiv2026

The unlabeled list color function of disconnected graphs

Given a graph $G$, its chromatic polynomial $P (G, k)$ counts proper $k$-colorings, while the corresponding list color function $P_{\ell} (G, k)$ counts the minimum number of proper colorings across all assignments of $k$ colors to each vertex. While it is clear that $P_{\ell} (G, k) \leq P (G, k)$, Donner showed in 1992 that $P_{\ell} (G, k) = P (G, k)$ whenever $k$ is sufficiently large. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph. A list version of Hanlon's notion was introduced in 2024 by Kaul and Mudrock, who further raised the question of whether the analog of Donner's result holds in the unlabeled case. While they proved this for all connected point-determining graphs, even the case of the edgeless graph on $n$ vertices remained open and was posed as a conjecture. We prove this conjecture and show that it implies that, more generally, a disconnected graph satisfies the unlabeled analog of Donner's result if all of its connected components do.

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arXiv2026

A Spectral Turán Problem for a Fixed Tree

We study the spectral Turán problem for trees. To avoid limiting our perspective to specific families of trees, we parametrize trees in terms of their unique bipartition. We say $T \in \mathcal{T}_{m,l+1}^δ$ if $T$ is a tree of order $m$, where the order of the smaller partite set $A$ of $T$ is $l+1$, and $δ$ is the minimum degree of the vertices in $A$. The motivation for this parametrization comes from the recent proof of the spectral Erdős-Sós conjecture. For a given fixed tree $T$, we describe $\mathrm{SPEX}(n,T)$ and consequently, bound $\mathrm{spex}(n,T)$ in terms of $m,l,δ$ for that tree. Our approach combines spectral arguments with new results and constructions on embedding a tree $T \in \mathcal{T}_{m,l+1}^δ$ into graphs of the form $\overline{K}_l \vee m S_δ$. We give bounds on $\mathrm{spex}(n,T)$ within an error of $Θ(n^{-1/2})$ and $Θ(n^{-1})$ that are based on our embedding results for the given $T$.

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arXiv2026

On strongly and robustly critical graphs

In extremal combinatorics, it is common to focus on structures that are minimal with respect to a certain property. In particular, critical and list-critical graphs occupy a prominent place in graph coloring theory. Stiebitz, Tuza, and Voigt introduced strongly critical graphs, i.e., graphs that are $k$-critical yet $L$-colorable with respect to every non-constant assignment $L$ of lists of size $k-1$. Here we strengthen this notion and extend it to the framework of DP-coloring (or correspondence coloring) by defining robustly $k$-critical graphs as those that are not $(k-1)$-DP-colorable, but only due to the fact that $χ(G) = k$. We then seek general methods for constructing robustly critical graphs. Our main result is that if $G$ is a critical graph (with respect to ordinary coloring), then the join of $G$ with a sufficiently large clique is robustly critical; this is new even for strong criticality.

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arXiv2025

List Coloring the Cartesian Product of a Complete Graph and Complete Bipartite Graph

We study the list chromatic number of the Cartesian product of a complete graph of order $n$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $χ_{\ell}(K_n \square K_{a,b})$. At the 2024 Sparse Graphs Coalition's Workshop on algebraic, extremal, and structural methods and problems in graph colouring, Mudrock presented the following question: For each positive integer $a$, does $χ_{\ell}(K_n \square K_{a,b}) = n+a$ if and only if $b \geq (n+a-1)!^a/(a-1)!^a$? In this paper, we show the answer to this question is yes by studying $χ_{\ell}(H \square K_{a,b})$ when $H$ is strongly chromatic-choosable (a special form of vertex criticality) with the help of the list color function and analytic inequalities such as that of Karamata. Our result can be viewed as a generalization of the well-known result that $χ_{\ell}(K_{a,b}) = 1+a$ if and only if $b \geq a^a$.

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arXiv2025

Shameful Inequalities for List and DP Coloring of Graphs

The chromatic polynomial of a graph is an important notion in algebraic combinatorics that was introduced by Birkhoff in 1912; denoted $P(G,k)$, it equals the number of proper $k$-colorings of graph $G$. Enumerative analogues of the chromatic polynomial of a graph have been introduced for two well-studied generalizations of ordinary coloring, namely, list colorings: $P_{\ell}$, the list color function (1990); and DP colorings: $P_{DP}$, the DP color function (2019), and $P^*_{DP}$, the dual DP color function (2021). For any graph $G$ and $k \in \mathbb{N}$, $P_{DP}(G, k) \leq P_\ell(G,k) \leq P(G,k) \leq P_{DP}^*(G,k)$. In 2000, Dong settled a conjecture of Bartels and Welsh from 1995 known as the Shameful Conjecture by proving that for any $n$-vertex graph $G$, $P(G,k+1)/(k+1)^n \geq P(G,k)/k^n$ for all $k \in \mathbb{N}$ satisfying $k \geq n-1$. In contrast, for infinitely many positive integers $n$, Seymour (1997) gave an example of an $n$-vertex graph for which the above inequality does not hold for some $k = Θ(n/ \log n)$. In this paper, we consider analogues of Dong's result for list and DP color functions. Specifically, in contrast to the chromatic polynomial, we prove that for any $n$-vertex graph $G$, $P_{\ell}(G,k+1)/(k+1)^n \geq P_{\ell}(G,k)/k^n$ and $P_{DP}(G,k+1)/(k+1)^n \geq P_{DP}(G,k)/k^n$ for all $k \in \mathbb{N}$. For the dual DP analogue of these inequalities, we show that there is a graph $G$ and $k \in \mathbb{N}$ such that $P_{DP}^*(G,k+1)/(k+1)^n < P_{DP}^*(G,k)/k^n$, and we prove $P_{DP}^*(G,k+1)/(k+1)^n \geq P_{DP}^*(G,k)/k^n$ for all $k \in \mathbb{N}$ satisfying $k \geq n-1$ when $G$ is an $n$-vertex complete bipartite graph.

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arXiv2025

On the DP-chromatic Number of Cartesian Products of Critical Graphs

DP-coloring (also called correspondence coloring) is a well-studied generalization of list coloring introduced by Dvořák and Postle in 2015. The following sharp bound on the DP-chromatic number of the Cartesian product of graphs $G$ and $H$ is known: $χ_{DP}(G \square H) \leq \text{min}\{χ_{DP}(G) + \text{col}(H), χ_{DP}(H) + \text{col}(G) \} - 1$ where $χ_{DP}(G)$ is the DP-chromatic number of $G$ and $\text{col}(H)$ is the coloring number of $H$. We seek to understand when $χ_{DP}(G \square K_{l,t})$ is far from its chromatic number: $χ(G \square K_{l,t}) = \max \{χ(G), 2 \}$ in the case that $G$ is a $k$-critical graph with $χ_{DP}(G)=k$. In particular, we have $χ_{DP}(G \square K_{l,t}) \leq k + l$, and for fixed $l$ we wish to find the smallest $t$ for which this upper bound is achieved. This can be viewed as an extension of the classic result that the list chromatic number of $K_{l,t}$ is $l+1$ if and only if $t \geq l^l$. Our results illustrate that the DP color function of $G$, the DP analogue of the chromatic polynomial, provides a concept and tool that is useful for making progress on this problem.

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arXiv2025

Counting Packings of List-colorings of Graphs

Given a list assignment for a graph, list packing asks for the existence of multiple pairwise disjoint list colorings of the graph. Several papers have recently appeared that study the existence of such a packing of list colorings. Formally, a proper $L$-packing of size $k$ of a graph $G$ is a set of $k$ pairwise disjoint proper $L$-colorings of $G$ where $L$ is a list assignment of colors to the vertices of $G$. In this note, we initiate the study of counting such packings of list colorings of a graph. We define $P_\ell^\star(G,q,k)$ as the guaranteed number of proper $L$-packings of size $k$ of $G$ over all list assignments $L$ that assign $q$ colors to each vertex of $G$, and we let $P^\star(G,q,k)$ be its classical coloring counterpart. We let $P_\ell^\star(G,q)= P_\ell^\star(G,q,q)$ so that $P_\ell^\star(G,q)$ is the enumerative function for the previously studied list packing number $χ_\ell^\star(G)$. Note that the chromatic polynomial of $G$, $P(G,q)$, is $P^\star(G,q,1)$, and the list color function of $G$, $P_\ell(G,q)$, is $P_\ell^\star(G,q,1)$. Inspired by the well-known behavior of the list color function and the chromatic polynomial, we make progress towards the question of whether $P_{\ell}^\star(G,q,k) = P^\star(G,q,k)$ when $q$ is large enough. Our result generalizes the recent theorem of Dong and Zhang (2023), which improved results going back to Donner (1992), about when the list color function equals the chromatic polynomial. Further, we use a polynomial method to generalize bounds on the list packing number, $χ_\ell^\star(G)$, of sparse graphs to exponential lower bounds (in the number of vertices of $G$) on the corresponding list packing functions, $P_\ell^\star(G,q)$.

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arXiv2025

DP-Coloring of Graphs from Random Covers

DP-coloring (also called correspondence coloring) of graphs is a generalization of list coloring that has been widely studied since its introduction by Dvořák and Postle in $2015$. Intuitively, DP-coloring generalizes list coloring by allowing the colors that are identified as the same to vary from edge to edge. Formally, DP-coloring of a graph $G$ is equivalent to an independent transversal in an auxiliary structure called a DP-cover of $G$. In this paper, we introduce the notion of random DP-covers and study the behavior of DP-coloring from such random covers. We prove a series of results about the probability that a graph is or is not DP-colorable from a random cover. These results support the following threshold behavior on random $k$-fold DP-covers as $ρ\to\infty$ where $ρ$ is the maximum density of a graph: graphs are non-DP-colorable with high probability when $k$ is sufficiently smaller than $ρ/\lnρ$, and graphs are DP-colorable with high probability when $k$ is sufficiently larger than $ρ/\lnρ$. Our results depend on $ρ$ growing fast enough and imply a sharp threshold for dense enough graphs. For sparser graphs, we analyze DP-colorability in terms of degeneracy. We also prove fractional DP-coloring analogs to these results.

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arXiv2024

A Polynomial Method for Counting Colorings of Sparse Graphs

The notion of $S$-labeling of graphs, where $S$ is a subset of a symmetric group, was introduced in 2019 by Jin, Wong, and Zhu. This notion provides the framework for a common generalization of various well studied notions of graph coloring, including classical coloring, signed $k$-coloring, signed $\mathbb{Z}_k$-coloring, DP (or correspondence) coloring, group coloring, and coloring of gained graphs. In this paper, we present a unified and simple polynomial method for giving exponential lower bounds on the number of colorings of an $S$-labeled graph for all such $S$. This algebraic technique allows us to prove new lower bounds on the number of colorings of any $S$-labeling of graphs satisfying certain sparsity conditions. We also investigate how the structure of $S$ can be exploited to improve the applicability of these bounds. Our results give new lower bounds on the number of DP-colorings, and consequently the number of all types of colorings listed above. This includes the chromatic polynomial and the number of list colorings of families of planar graphs, and the number of colorings of signed graphs. These enumerative bounds improve previously known results or are the first such known results.

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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

Longitudinal Network Models and Permutation-Uniform Markov Chains

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.

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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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