arXiv subjects
Publications and source records attributed to Hemanshu Kaul.
We study the NP-complete Maximum Outerplanar Subgraph problem. The previous best known approximation ratio for this problem is 2/3. We propose a new approximation algorithm which improves the ratio to 7/10.
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.
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 $τ(K_{2,4}) = τ(K_{2,5}) = 3$.
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 $χ_{DP}(G \square K_{k,t})= χ_{DP}(G) + k$?
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. Chromatic polynomials for joins and vertex-gluings of graphs are well understood, but the effect of these graph operations on the DP color function is not known. In this paper we make progress on understanding the DP color function of the join of a graph with a complete graph and vertex-gluings of certain graphs. We also develop tools to study the DP color function under these graph operations, and we study the threshold (smallest $m$) beyond which the DP color function of a graph constructed with these operations equals its chromatic polynomial.
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 $k$.
We prove a common generalization of the celebrated Sauer-Spencer packing theorem and a theorem of Brandt concerning finding a copy of a tree inside a graph. This proof leads to the characterization of the extremal graphs in the case of Brandt's theorem: If $G$ is a graph and $F$ is a forest, both on $n$ vertices, and $3Δ(G)+\ell^*(F)\leq n$, then $G$ and $F$ pack unless $n$ is even, $G=\frac{n}{2}K_2$ and $F=K_{1,n-1}$; where $\ell^*(F)$ is the difference between the number of leaves and twice the number of nontrivial components of $F$.
We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its {computationally} efficient solution.
We present a flexible public transit network design model which optimizes a social access objective while guaranteeing that the system's costs and transit times remain within a preset margin of their current levels. The purpose of the model is to find a set of minor, immediate modifications to an existing bus network that can give more communities access to the chosen services while having a minimal impact on the current network's operator costs and user costs. Design decisions consist of reallocation of existing resources in order to adjust line frequencies and capacities. We present a hybrid tabu search/simulated annealing algorithm for the solution of this optimization-based model. As a case study we apply the model to the problem of improving equity of access to primary health care facilities in the Chicago metropolitan area. The results of the model suggest that it is possible to achieve better primary care access equity through reassignment of existing buses and implementation of express runs, while leaving overall service levels relatively unaffected.
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.
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).
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.
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.
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 Dvořák 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 $m$.
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 $χ_\ell(G \square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $χ_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since $χ_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells us the values of $b$ for which $χ_\ell(K_{a,b})$ is as large as possible and far from $χ(K_{a,b})=2$. In this paper we seek to understand when $χ_\ell(G \square K_{a,b})$ is far from $χ(G \square K_{a,b}) = \max \{χ(G), 2 \}$. It is easy to show $χ_\ell(G \square K_{a,b}) \leq χ_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Král, and Miskuf showed that this bound is attainable if $b$ is sufficiently large; specifically, $χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$ whenever $b \geq (χ_\ell(G) + a - 1)^{a|V(G)|}$. Given any graph $G$ and $a \in \mathbb{N}$, we wish to determine the smallest $b$ such that $χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$. 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.
We introduce a notion of color-criticality in the context of chromatic-choosability. We define a graph $G$ to be strong $k$-chromatic-choosable if $χ(G) = k$ and every $(k-1)$-assignment for which $G$ is not list-colorable has the property that the lists are the same for all vertices. That is the usual coloring is, in some sense, the obstacle to list-coloring. We prove basic properties of strongly chromatic-choosable graphs such as chromatic-choosability and vertex-criticality, and we construct infinite families of strongly chromatic-choosable graphs. We derive a sufficient condition for the existence of at least two list colorings of strongly chromatic-choosable graphs and use it to show that: if $M$ is a strong $k$-chromatic-choosable graph with $|E(M)| \leq |V(M)|(k-2)$ and $H$ is a graph that contains a Hamilton path, $w_1, w_2, \ldots, w_m$, such that $w_i$ has at most $ρ\geq 1$ neighbors among $w_1, \ldots, w_{i-1}$, then $χ_{\ell}(M \square H) \le k+ ρ- 1$. We show that this bound is sharp for all $ρ\ge 1$ by generalizing the theorem to apply to $H$ that are $(M,ρ)$-Cartesian accommodating which is a notion we define with the help of the list color function, $ P_{\ell}(G,k)$, the list analogue of the chromatic polynomial. We also use the list color function to determine the list chromatic number of certain star-like graphs: $χ_{\ell}(M \square K_{1,s}) =$ $k \; \text{if } s < P_{\ell}(M,k)$, or $k+1 \; \text{if } s \geq P_{\ell}(M,k)$, where $M$ is a strong $k$-chromatic-choosable graph. We show that $ P_{\ell}(M,k)$ equals $P(M,k)$, the chromatic polynomial, when $M$ is an odd cycle, complete graph, or the join of an odd cycle with a complete graph.
We recently introduced proportional choosability, a new list analogue of equitable coloring. Like equitable coloring, and unlike list equitable coloring (a.k.a. equitable choosability), proportional choosability bounds sizes of color classes both from above and from below. In this note, we show that a graph is proportionally 2-choosable if and only if it is a linear forest such that its largest component has at most 5 vertices and all of its other components have two or fewer vertices. We also construct examples that show that characterizing equitably 2-choosable graphs is still open.
In 2003, Kostochka, Pelsmajer, and West introduced a list analogue of equitable coloring called equitable choosability. In this paper, we motivate and define a new list analogue of equitable coloring called proportional choosability. A $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors for each vertex $v$ of $G$. An $L$-coloring assigns a color to each vertex $v$ from its list $L(v)$. For each color $c$, let $η(c)$ be the number of vertices $v$ whose list $L(v)$ contains $c$. A proportional $L$-coloring of $G$ is a proper $L$-coloring in which each color $c \in \bigcup_{v \in V(G)} L(v)$ is used $\lfloor η(c)/k \rfloor$ or $\lceil η(c)/k \rceil$ times. A graph $G$ is proportionally $k$-choosable if a proportional $L$-coloring of $G$ exists whenever $L$ is a $k$-assignment for $G$. We show that if a graph $G$ is proportionally $k$-choosable, then every subgraph of $G$ is also proportionally $k$-choosable and also $G$ is proportionally $(k+1)$-choosable, unlike equitable choosability for which analogous claims would be false. We also show that any graph $G$ is proportionally $k$-choosable whenever $k \geq Δ(G) + \lceil |V(G)|/2 \rceil$, and we use matching theory to completely characterize the proportional choosability of stars and the disjoint union of cliques.