A graph $G$ is called chromatic-choosable if $χ(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are $k$-choosable. This upper bound on $|V(G)|$ is tight. It is known that if $k$ is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $|V(G)| =2 k+2$ that are not $k$-choosable. Some subgraphs of these two graphs are also non-$k$-choosable. The main result of this paper is that all other $k$-chromatic graphs $G$ with $|V(G)| =2 k+2$ are $k$-choosable. In particular, if $χ(G)$ is odd and $|V(G)| \le 2χ(G)+2$, then $G$ is chromatic-choosable, which was conjectured by Noel.
It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are chromatic-choosable. This upper bound on $|V(G)|$ is tight: if $k$ is even, then $K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $2 k+2$ vertices that are not chromatic-choosable. It was proved in [arXiv:2201.02060] that these are the only non-$k$-choosable complete $k$-partite graphs with $2k+2$ vertices. For $G =K_{3 \star (k/2+1), 1 \star (k/2-1)}$ or $K_{4, 2 \star (k-1)}$, a bad list assignment of $G$ is a $k$-list assignment …
Cones of Theta Graphs are Enumeratively Chromatic-Choosable
School of Computing, University of South Alabama（南阿拉巴马大学计算机学院）；McGill-Toolen Catholic High School（麦吉尔-图伦天主教高中）；Department of Mathematics and Statistics, University of South Alabama（南阿拉巴马大学数学与统计系）；Department of Applied Mathematics, Illinois Institute of Technology（伊利诺伊理工学院应用数学系）；Alabama School of Math and Science（阿拉巴马数学与科学学校）；Department of Biomedical Sciences, University of South Alabama（南阿拉巴马大学生物医学科学系）
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_{\ell}(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_{\ell} (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Motivated by the List Total Coloring Conjecture, Kostochka and Woodall (2001) proposed the List Square Coloring Conjecture which states that $G^2$ is chromatic-choosable …
Among other results, it is shown that 3-trees are $Δ$-edge-choosable and that graphs of tree-width 3 and maximum degree at least 7 are $Δ$-edge-choosable.
The $k$th power $G^k$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^k$ if the distance between $u$ and $v$ in $G$ is at most $k$. Let $χ(H)$ and $χ_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_l (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. A natural question raised by Xuding Zhu (2012) is whether there exists a constant integer $k$ such that $G^k$ is chromatic-choosable for every graph $G$. Motivated by the Li…
For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a list assignment $L$ of $G$ such that the colour set $\bigcup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $q$ sets so that for each $i$ and each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if $G$ is $L$-colourable for any $λ$-list assignment $L$ of $G$. The concept of $λ$-choosability puts $k$-colourability and $k$-choosability in the same framework: If $λ= \{k\}$, then $λ$-choosab…
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 e…
Let $Σ$ be a signed graph where two edges joining the same pair of vertices with opposite signs are allowed. The zero-free chromatic number $χ^*(Σ)$ of $Σ$ is the minimum even integer $2k$ such that $G$ admits a proper coloring $f\colon\,V(Σ)\mapsto \{\pm 1,\pm 2,\ldots,\pm k\}$. The zero-free list chromatic number $χ^*_l(Σ)$ is the list version of zero-free chromatic number. $Σ$ is called zero-free chromatic-choosable if $χ^*_l(Σ)=χ^*(Σ)$. We show that if $Σ$ has at most $χ^*(Σ)+1$ vertices then $Σ$ is zero-free chromatic-choosable. This result strengthens Noel-Reed-Wu Theorem which states th…
Let $ϕ(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $ϕ(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is known that $ϕ(k) \le 2k+3$ and it is conjectured by Noel that $ϕ(k) = 2k+3$. For a multi-set $λ=\{k_1,k_2, \ldots, k_q\}$ of positive integers, let $k_λ = \sum_{i=1}^q k_i$. A $λ$-list assignment of $G$ is a $k_λ$-list assignment $L$ for which the colour set $\cup_{v \in V(G)}L(v)$ can be partitioned into the disjoint union $C_1 \cup C_2 \cup \ldots \cup C_q$ of $…
We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $Δ$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $Δ\ge 6$, if $\mad(G)<2+\frac{4Δ-8}{5Δ+2}$, then $\chil(G^2)=Δ+1$. In particular, if $G$ is planar with girth $g\ge 7+\frac{12}{Δ-2}$, then $\chil(G^2)=Δ+1$. Under the same conditions, $\chil^i(G)=Δ$, where $\chil^i$ is the list injective chromatic number.
The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two vertices $u$ and $v$ are adjacent in $G^2$ if the distance between $u$ and $v$ in $G$ is at most 2. Let $χ(H)$ and $χ_l(H)$ be the chromatic number and the list chromatic number of $H$, respectively. A graph $H$ is called {\em chromatic-choosable} if $χ_l (H) = χ(H)$. It is an interesting problem to find graphs that are chromatic-choosable. Kostochka and Woodall \cite{KW2001} conjectured that $χ_l(G^2) = χ(G^2)$ for every graph $G$, which is called List Square Coloring Conjecture. In this paper, we give infinitely man…
Recently, Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $χ(G^k)$ unbounded and $χ_{\ell}(G^k)\geq c χ(G^k) \log χ(G^k)$. We also provide an upper bound, $χ_{\ell}(G^k)<χ(G^k)^3$ for $k>1$.
A wheel graph consists of a cycle along with a center vertex connected to every vertex in the cycle. In this paper we show that every subgraph of a wheel graph has list coupled chromatic number at most 5, and this coloring can be found in linear time. We further show that `5' is tight for every wheel graph with at least 5 vertices, and briefly discuss possible generalizations to planar graphs of treewidth 3.
A graph $G$ is total weight $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a proper total $L$-weighting, i.e., a mapping $f: V(G) \cup E(G) \to \mathbb{R}$ such that for each $z \in V(G) \cup E(G)$, $f(z) \in L(z)$, and for each edge $uv$ of $G$, $\sum_{e \in E(u)}f(e)+f(u) \ne \sum_{e \in E(v)}f(e) + f(v)$. This paper proves that if $G$ decomposes into complete graphs of odd order, then $G$ is total weight $(1,3)$-choosable. As a consequence, every Eulerian…
This paper studies the on-line choice number on complete multipartite graphs with independence number $m$. We give a unified strategy for every prescribed $m$. Our main result leads to several interesting consequences comparable to known results. (1) If $ k_1-\sum_{p=2}^m(\frac{p^2}{2}-\frac{3p}{2}+1)k_p\geq 0$, where $k_p$ denotes the number of parts of cardinality $p$, then $G$ is on-line chromatic-choosable. (2) If $ |V(G)|\leq\frac{m^2-m+2}{m^2-3m+4}χ(G)$, then $G$ is on-line chromatic-choosable. (3) The on-line choice number of regular complete multipartite graphs $K_{m\star k}$ is at mos…
Let $G$ be a graph on $n$ vertices and let $\mathcal{L}_k$ be an arbitrary function that assigns each vertex in $G$ a list of $k$ colours. Then $G$ is $\mathcal{L}_k$-list colourable if there exists a proper colouring of the vertices of $G$ such that every vertex is coloured with a colour from its own list. We say $G$ is $k$-choosable if for every such function $\mathcal{L}_k$, $G$ is $\mathcal{L}_k$-list colourable. The minimum $k$ such that $G$ is $k$-choosable is called the list chromatic number of $G$ and is denoted by $χ_L(G)$. Let $χ_L(G) = s$ and let $t$ be a positive integer less than …
A signed graph is a graph in which each edge is labeled with $+1$ or $-1$. A (proper) vertex coloring of a signed graph is a mapping $\f$ that assigns to each vertex $v\in V(G)$ a color $\f(v)\in \mz$ such that every edge $vw$ of $G$ satisfies $\f(v)\not= \sg(vw)\f(w)$, where $\sg(vw)$ is the sign of the edge $vw$. For an integer $h\geq 0$, let $\Ga_{2h}=\{\pm1,\pm2, \ldots, \pm h\}$ and $\Ga_{2h+1}=\Ga_{2h} \cup \{0\}$. Following \cite{MaRS2015}, the signed chromatic number $\scn(G)$ of $G$ is the least integer $k$ such that $G$ admits a vertex coloring $\f$ with ${\rm im}(\f)\subseteq \Ga_k$…
We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $Δ(G) \geq 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(mΔ(G):m)$-edge-choosable and $Δ(G)$-edge-paintable for every $m \geq 1$.
It was conjectured by Ohba and confirmed recently by Noel et al. that, for any graph $G$, if $|V(G)|\le 2χ(G)+1$ then $χ_l(G)=χ(G)$. This indicates that the graphs with high chromatic number are chromatic-choosable. We show that this is also the case for uniform hypergraphs and further propose a generalized version of Ohba's conjecture: for any $r$-uniform hypergraph $H$ with $r\geq 2$, if $|V(H)|\le rχ(H)+r-1$ then $χ_l(H)=χ(H)$. We show that the condition of the proposed conjecture is sharp by giving two classes of $r$-uniform hypergraphs $H$ with $|V(H)|= rχ(H)+r$ and $χ_l(H)>χ(H)$. To supp…
A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=χ(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2χ(G)+1$ is chromatic-choosable. Since each $k$-chromatic graph is a subgraph of a complete $k$-partite graph, we see that Ohba's conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba's conjecture has been verified are: $K_{3*2,2*(k-3),1}$, $K_{3,2*(k-1)}$, $K_{s+3,2*(k-s-1),1*s}$, $K_{4,3,2*(k-4),1*2}$, and $K_{5,3,…
The choosability $χ_\ell(G)$ of a graph $G$ is the minimum $k$ such that having $k$ colors available at each vertex guarantees a proper coloring. Given a toroidal graph $G$, it is known that $χ_\ell(G)\leq 7$, and $χ_\ell(G)=7$ if and only if $G$ contains $K_7$. Cai, Wang, and Zhu proved that a toroidal graph $G$ without 7-cycles is 6-choosable, and $χ_\ell(G)=6$ if and only if $G$ contains $K_6$. They also prove that a toroidal graph $G$ without 6-cycles is 5-choosable, and conjecture that $χ_\ell(G)=5$ if and only if $G$ contains $K_5$. We disprove this conjecture by constructing an infinite…
Let $Q(n,c)$ denote the minimum clique number over graphs with $n$ vertices and chromatic number $c$. We determine the rate of growth of of the sequence ${Q(n,\lceil rn \rceil)}_{n=1}^\infty$ for any fixed $0<r\leq 1$. We also give a better upper bound for $Q(n,\lceil rn \rceil)$.
Let $f:V \rightarrow \mathbb{N}$ be a function on the vertex set of the graph $G=(V,E)$. The graph $G$ is {\em $f$-choosable} if for every collection of lists with list sizes specified by $f$ there is a proper coloring using colors from the lists. The sum choice number, $χ_{sc}(G)$, is the minimum of $\sum f(v)$, over all functions $f$ such that $G$ is $f$-choosable. It is known (Alon 1993, 2000) that if $G$ has average degree $d$, then the usual choice number $χ_\ell(G)$ is at least $Ω(\log d)$, so they grow simultaneously. In this paper we show that $χ_{sc}(G)/|V(G)|$ can be bounded while th…
We study the list chromatic number of Cartesian products of graphs through the Alon-Tarsi number as defined by Jensen and Toft (1995) in their seminal book on graph coloring problems. The Alon-Tarsi number of $G$, $AT(G)$, is the smallest $k$ for which there is an orientation, $D$, of $G$ with max indegree $k\!-\!1$ such that the number of even and odd circulations contained in $D$ are different. It is known that $χ(G) \leq χ_\ell(G) \leq χ_p(G) \leq AT(G)$, where $χ(G)$ is the chromatic number, $χ_\ell(G)$ is the list chromatic number, and $χ_p(G)$ is the paint number of $G$. In this paper we…
For a graph $G$, the tree graph ${\cal T}_{G,t}$ has all tree subgraphs of $G$ with $t$ vertices as vertex set and two tree subgraphs are neighbors if they are edge-disjoint. Also, the $r^{th}$ cut number of $G$ is the minimum number of edges between parts of a partition of vertex set of $G$ into two parts such that each part has size at least $r$. We show that if $t=(1-o(1))n$ and $n$ is large enough, then for any dense graph $G$ with $n$ vertices, the chromatic number of the tree graph ${\cal T}_{G,t}$ is equal to the $(n-t+1)^{th}$ cut number of $G$. In particular, as a consequence, we prov…
A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $χ_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The equitable chromatic threshold of a graph $G$, denoted by $χ_=^*(G)$, is the minimum $t$ such that $G$ is equitably $k$-colorable for $k\ge t$. We develop a formula and a linear-time algorithm which compute the equitable chromatic threshold of an arbitrary complete multipartite graph.
Let $G$ be a simple graph. The dominated coloring of $G$ is a proper coloring of $G$ such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of $G$ is called the dominated chromatic number of $G$, denoted by $χ_{dom}(G)$. Stability (bondage number) of dominated chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the dominated chromatic number of $G$. In this paper, we study the dominated chromatic number, dominated stability and dominated bondage number of certain graphs.
Let $G$ be a plane graph. A vertex-colouring $φ$ of $G$ is called {\em facial non-repetitive} if for no sequence $r_1 r_2 \dots r_{2n}$, $n\geq 1$, of consecutive vertex colours of any facial path it holds $r_i=r_{n+i}$ for all $i=1,2,\dots,n$. A plane graph $G$ is {\em facial non-repetitively $l$-choosable} if for every list assignment $L:V\rightarrow 2\sp{\mathbb{N}}$ with minimum list size at least $l$ there is a facial non-repetitive vertex-colouring $φ$ with colours from the associated lists. The {\em facial Thue choice number}, $π_{fl}(G)$, of a plane graph $G$ is the minimum number $l$ …
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_…
The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree with strength k that has maximum degree only 2k-2. The result is best possible.
A number of famous results about triangle-free graphs determine the maximum chromatic number of graphs of order n with minimum degree greater than n/3. In this note these results are extended to Kr-free graphs.
A vertex coloring of a given graph $G$ is conflict-free if the closed neighborhood of every vertex contains a unique color (i.e. a color appearing only once in the neighborhood). The minimum number of colors in such a coloring is the conflict-free chromatic number of $G$, denoted $χ_{CF}(G)$. What is the maximum possible conflict-free chromatic number of a graph with a given maximum degree $Δ$? Trivially, $χ_{CF}(G)\leq χ(G)\leq Δ+1$, but it is far from optimal - due to results of Glebov, Szabó and Tardos, and of Bhyravarapu, Kalyanasundaram and Mathew, the answer in known to be $Θ\left(\ln^2Δ…
The (independent) chromatic vertex stability ($\ivs(G)$) $\vs(G)$ is the minimum size of (independent) set $S\subseteq V(G)$ such that $χ(G-S)=χ(G)-1$. In this paper we construct infinitely many graphs $G$ with $Δ(G)=4$, $χ(G)=3$, $\ivs(G)=3$ and $\vs(G)=2$, which gives a partial negative answer to a problem posed in \cite{ABKM}.
Let Q(n,c) denote the minimum clique size an n-vertex graph can have if its chromatic number is c. Using Ramsey graphs we give an exact, albeit implicit, formula for the case c is at least (n+3)/2.
In this paper we obtain some upper bounds for $b$-chromatic number of $K_{1,t}$ -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight.
Let G=(V,E) be a graph and let f be a function that assigns list sizes to the vertices of G. It is said that G is f-choosable if for every assignment of lists of colors to the vertices of G for which the list sizes agree with f, there exists a proper coloring of G from the lists. The sum choice number is the minimum of the sum of list sizes for f over all choosable functions f for G. The sum choice number of a graph is always at most the sum |V|+|E|. When the sum choice number of G is equal to this upper bound, G is said to be sc-greedy. In this paper, we determine the sum choice number of all…
Mathematics Institute, University of Warwick（华威大学数学研究所）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极值组合与概率研究组）；Departament de Matemàtiques, Universitat Politècnica de Catalunya (UPC)（加泰罗尼亚理工大学数学系）；School of Mathematics and Statistics, Ningbo University（宁波大学数学与统计学院）
Two subgraphs $A,B$ of a graph $G$ are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if $G$ is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results. We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have …
