We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we revisit their ideas and reinterpret their equivalence relation in terms of a new quotient algebra of NCSym. We investigate the projection of the chromatic symmetric function $Y_G$ in noncommuting variables in this quotient algebra, which defines $y_{G : v}$, the chromatic symmetric function of a graph G centred at a vertex v. We then apply our methods to $y_{G :v}$ and find new families of unit interval graphs that are (e)-positive, a stronger condition than classical e-positivity, thus confirming new cases of the (3+1)-free conjecture of Stanley and Stembridge.
In our study of $y_{G : v}$, we also describe methods of constructing new e-positive graphs from given $(e)$-positive graphs and classify the (e)-positivity of trees and cut vertices. We moreover construct a related quotient algebra of NCQSym to prove theorems relating the coefficients of $y_{G : v}$ to acyclic orientations of graphs, including a noncommutative refinement of Stanley's sink theorem.
The $b$-Chromatic Number and $f$-Chromatic Vertex Number of Regular Graphs
图$G$的$b$-色数，记为$b(G)$，是使G存在一个$k$色的真染色的最大正整数，其中每个色类至少包含一个与其它色类中某个顶点相邻的顶点，这样的顶点称为优势点.一个$d$-正则图$G$的$f$-色顶点数记为$f(G)$，是在一个含有$d+1$颜色的真染色中，不同颜色的主顶点的最大数目. El Sahili和Kouider证明了对任意周长为5的$d$-正则图$G$，都有$b(G)=d+1$.在补充条件下，我们研究了这一猜想，给出了部分答案。
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}.
We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally ad…
A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the graph has a b-coloring with k colors. In this work we present lower bounds for the b-chromatic number of a vertex-deleted subgraph of a graph, particularly regarding two important classes, quasi-line and chordal graphs. We also get bounds for the b-chromatic number of G -{x}, when G is a graph with large girth.
For an arbitrary invariant $ρ(G)$ of a graph $G$, the $ρ-$vertex stability number $vs_ρ(G)$ is the minimum number of vertices of $G$ whose removal results in a graph $H\subseteq G$ with $ρ(H)\neq ρ(G)$ or with $E(H)=\varnothing$. In this paper, first we give some general lower and upper bounds for the $ρ$-vertex stability number, and then study the edge chromatic stability number of graphs, $vs_{χ^{\prime}}(G)$, where $χ^{\prime}=χ^{\prime}(G)$ is edge chromatic number (chromatic index) of $G$. We prove some general results for this parameter and determine $vs_{χ^{\prime}}(G)$ for specific cla…
The chromatic vertex (resp.\ edge) stability number ${\rm vs}_χ(G)$ (resp.\ ${\rm es}_χ(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $χ(H)=χ(G)-1$. In the main result it is proved that if $G$ is a graph with $χ(G) \in \{ Δ(G), Δ(G)+1 \}$, then ${\rm vs}_χ(G) = {\rm ivs}_χ(G)$, where ${\rm ivs}_χ(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $χ(G) \le \frac{Δ(G)+1}{2}$. It is proved that if $χ(G) > \frac{Δ(G)}{2}+1$, then ${\rm vs}_χ(G) = {\rm es}_χ(G)$. A Nordhaus-Gaddum-…
Several games that arise from graph coloring have been introduced and studied. Let $φ$ denote a graph invariant that arises from such a game. If $G$ is a graph and $φ(G-x)\neq φ(G)=k$, $k \geq 1$, holds true for every vertex $x \in V(G)$, then $G$ is called a $k$-$φ$-game-vertex-critical graph. We study the concept of $φ$-game-vertex-criticality for $φ\in \{χ_g, χ_i, χ_{ig}^{A}, χ_{ig}^{AB}\}$, where $χ_g$ denotes the standard game chromatic number, $χ_i$ denotes the indicated game chromatic number and $χ_{ig}^{A}$, $χ_{ig}^{AB}$ denote two versions of the independence game chromatic number. S…
This paper has two main parts. First, we consider the Tutte symmetric function $XB$, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of $XB$ and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted $XB$ admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting $XB$ to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific e…
In this paper, we prove that every 3-chromatic connected graph, except $C_7$, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path. It is a special case of a conjecture due to S.~Akbari, F.~Khaghanpoor, and S.~Moazzeni, cited in [P.J. Cameron, Research problems from the BCC22, \emph{Discrete Math.} {\bf 311} (2011), 1074--1083], stating that every connected graph $G$ other than $C_7$ admits a $χ(G)$-coloring such that every vertex of $G$ is the beginning of a colorful path (i.e. a path of on $χ(G)$ vertices containing a vertex of each color). We also provide …
Given a graph $G$, a function $c:V(G)\longrightarrow \{1,\ldots,k\}$ with the property that $c(u)=c(v)=i$ implies that the distance between $u$ and $v$ is greater than $i$, is called a $k$-packing coloring of $G$. The smallest integer $k$ for which there exists a $k$-packing coloring of $G$ is called the packing chromatic number of $G$, and is denoted by $χ_ρ$. Packing chromatic vertex-critical graphs are the graphs $G$ for which $χ_ρ(G-x) < χ_ρ(G)$ holds for every vertex $x$ of $G$. A graph $G$ is called a packing chromatic critical graph if for every proper subgraph $H$ of $G$, $χ_ρ(H) < χ_ρ…
Let $G$ be a connected undirected graph.~A vertex coloring $f$ of $G$ is an $N_i$-vertex coloring if for each vertex $x$ in $G$, the number of different colors assigned to $N_G(x)$ is at most $i$.~The $N_i$-chromatic number of $G$, denoted by $t_i(G)$, is the maximum number of colors which are used in an $N_i$-vertex coloring of $G$. In this paper, we provide sharp bounds for $t_i(G)$ of a graph $G$ in terms of its vertex cover number, maximum degree and diameter, respectively. We also determine precise values for $t_i(G)$ in some cases.
A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $ω\ge 3$ has chromatic number at most $ω^{\log_2(ω)}$. The best previous result was an exponential upper bound $(5/27)3^ω$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.
The anagram-free chromatic number is a new graph parameter introduced independently Kamčev, Łuczak, and Sudakov (2017) and Wilson and Wood (2017). In this note, we show that there are planar graphs of pathwidth 3 with arbitrarily large anagram-free chromatic number. More specifically, we describe $2n$-vertex planar graphs of pathwidth 3 with anagram-free chromatic number $Ω(\log n)$. We also describe $kn$ vertex graphs with pathwidth $2k-1$ having anagram-free chromatic number in $Ω(k\log n)$.
Schrijver graphs are vertex-color-critical subgraphs of Kneser graphs having the same chromatic number. They also share the value of their fractional chromatic number but Schrijver graphs are not critical for that. Here we present an induced subgraph of every Schrijver graph that is vertex-critical with respect to the fractional chromatic number. These subgraphs turn out to be isomorphic with certain circular complete graphs. We also characterize the critical edges within this subgraph.
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we examine the effects on $χ_d^t(G)$ when $G$ is modified by operations on vertex and edge of $G$.
Let $G$ be a simple graph. A total dominator coloring of $G$ is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic number $χ_d^t(G)$ of $G$ is the minimum number of colors among all total dominator coloring of $G$. In this paper, we study the total dominator chromatic number of some specific graphs.
A generalized vertex join of a graph is obtained by joining an arbitrary multiset of its vertices to a new vertex. We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs. We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.
For a graph $G = (V(G), E(G))$, a dominating set $D$ is a vertex subset of $V(G)$ in which every vertex of $V(G) \setminus D$ is adjacent to a vertex in $D$. The domination number of $G$ is the minimum cardinality of a dominating set of $G$ and is denoted by $γ(G)$. A coloring of $G$ is a partition $C = (V_{1}, ... ,V_{k})$ such that each of $V_{i}$ in an independent set. The chromatic number is the smallest $k$ among all colorings $C = (V_{1}, ... ,V_{k})$ of $G$ and is denoted by $χ(G)$. A coloring $C = (V_{1}, ... ,V_{k})$ is said to be dominator if, for all $V_{i}$, every vertex $v \in V_{…
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In …
For graph $G$ and integers $a_1 \ge \cdots \ge a_r \ge 2$, we write $G \rightarrow (a_1 ,\cdots ,a_r)^v$ if and only if for every $r$-coloring of the vertex set $V(G)$ there exists a monochromatic $K_{a_i}$ in $G$ for some color $i \in \{1, \cdots, r\}$. The vertex Folkman number $F_v(a_1 ,\cdots ,a_r; s)$ is defined as the smallest integer $n$ for which there exists a $K_s$-free graph $G$ of order $n$ such that $G \rightarrow (a_1 ,\cdots ,a_r)^v$. It is well known that if $G \rightarrow (a_1 ,\cdots ,a_r)^v$ then $χ(G) \geq m$, where $m = 1+ \sum_{i=1}^r (a_i - 1)$. In this paper we study su…
We prove that for all nonnegative integers k,s there exists c with the following property. Let G be a graph with clique number at most k and chromatic number more than c. Then for every vertex-colouring (not necessarily optimal) of G, some induced subgraph of G is an s-vertex path, and all its vertices have different colours. This extends a recent result of Gyarfas and Sarkozy, who proved the same (when k=2) for graphs G with girth at least five.
We say that a vertex colouring $φ$ of a graph $G$ is nonrepetitive if there is no positive integer $n$ and a path on $2n$ vertices $v_{1}\ldots v_{2n}$ in $G$ such that the associated sequence of colours $φ(v_{1})\ldotsφ(v_{2n})$ satisfy $φ(v_{i})=φ(v_{i+n})$ for all $i=1,2,\dots,n$. The minimum number of colours in a nonrepetitive vertex colouring of $G$ is the Thue chromatic number $π(G)$. For the case of vertex list colourings the Thue choice number $π_{l}(G)$ of $G$ denotes the smallest integer $k$ such that for every list assignment $L:V(G)\rightarrow 2^{\mathbb{N}}$ with minimum list len…
Let $G$ be a simple graph. The dominated coloring of a graph $G$ 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 $χ_{dom}(G)$. In this paper, we examine the effects on $χ_{dom}(G)$ when $G$ is modified by operations on vertex and edge of $G$.
We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding $n$ leaves to a common vertex on an $m$-vertex cycle is not Schur positive if $m\ne 2n-1$, and conjecture that neither are the squid graphs…
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…
Hadwiger's famous coloring conjecture states that every $t$-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139--144] conjectured the following strengthening of Hadwiger's conjecture: If $G$ is a $t$-chromatic graph and $S \subseteq V(G)$ takes all colors in every $t$-coloring of $G$, then $G$ contains a $K_t$-minor rooted at $S$. We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for $5$-chromatic graphs as follows: Every $5$-chromatic graph contains a $K_5$-mi…
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.
Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $χ_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. For any $k \in \mathbb{N}$, the $k$-subdivision of $G$ is a simple graph $G^{\frac{1}{k}}$ which is constructed by replacing each edge of $G$ with a path of length $k$. In this paper, we study the total dominator chromatic number of $k$-subdivision of $G$.
For a simple graph $G$, a domination coloring of $G$ is a proper vertex coloring such that every vertex of $G$ dominates at least one color class, and every color class is dominated by at least one vertex. The domination chromatic number, denoted by $χ_{dd}(G)$, is minimum number of colors among all domination colorings of $G$. In this paper, we discuss the effects of some typical operations on $χ_{dd}(G)$, such as vertex (edge) removal, vertex (edge) contraction, edge subdivision, and cycle extending.
Assume $n, m$ are positive integers and $G$ is a graph. Let $P_{n,m}$ be the graph obtained from the path with vertices $\{-m, -(m-1), \ldots, 0, \ldots, n\}$ by adding a loop at vertex $ 0$. The double cone $Δ_{n,m}(G)$ over a graph $G$ is obtained from the direct product $G \times P_{n,m}$ by identifying $V(G) \times \{n\}$ into a single vertex $(\star, n)$, identifying $V(G) \times \{-m\}$ into a single vertex $(\star, -m)$, and adding an edge connecting $(\star, -m)$ and $(\star, n)$. This paper determines the fractional chromatic number of $Δ_{n,m}(G)$. In particular, if $n < m$ or $n=m$ …
Dominator coloring of a graph is a proper (vertex) coloring with the property that every vertex is either alone in its color class or adjacent to all vertices of at least one color class. A dominated coloring of a graph is a proper coloring such that every color class is dominated with at least one vertex. The dominator chromatic number of corona products and of edge corona products is determined. Sharp lower and upper bounds are given for the dominated chromatic number of edge corona products. The dominator chromatic number of hierarchical products is bounded from above and the dominated chro…
A class of graphs is $χ$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(ω(G))$, where $ω(G)$ is the clique number of $G$; the class is polynomially $χ$-bounded if $f$ can be taken to be a polynomial. The Gyárfás-Sumner conjecture asserts that, for every forest $H$, the class of $H$-free graphs (graphs with no induced copy of $H$) is $χ$-bounded. Let us say a forest $H$ is good if it satisfies the stronger property that the class of $H$-free graphs is polynomially $χ$-bounded. Very few forests are known to be good: for example, it is op…
University of West Bohemia（西波希米亚大学）；University of Maribor（马里博尔大学）；Institute of Mathematics, Physics and Mechanics（数学、物理与力学研究所）；University of Ljubljana（卢布尔雅那大学）
A note on vertex-critical induced subgraphs of shift graphs
University of West Bohemia（西波希米亚大学）；CNRS（法国国家科学研究中心）；IRIF（IRIF实验室）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学物理学院）；Faculty of Information Studies in Novo Mesto（诺沃梅斯托信息研究学院）；Rudolfovo – Science and Technology Centre（鲁多维福科学与技术中心）；University of Southern Denmark（南丹麦大学）
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…
An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label of $x$ is $f^+(x)= \sum_{e\in E(x)} f(e)$ ($E(x)$ is the set of edges incident to $x$). The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, several sufficient conditions to determine the local antimagic chromatic number of the join of graph…
The strong chromatic number $χ_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of spanning disjoint copies of $K_r$ in the same vertex set, the resulting graph has a proper vertex-colouring with $r$ colours. We show that for every $c > 0$ and every graph $G$ on $n$ vertices with $Δ(G) \ge cn$, $χ_{\text{s}}(G) \leq (2 + o(1)) Δ(G)$, which is asymptotically best possible.
A {\it local antimagic labeling} of a connected graph $G$ with at least three vertices, is a bijection $f:E(G) \rightarrow \{1,2,\ldots , |E(G)|\}$ such that for any two adjacent vertices $u$ and $v$ of $G$, the condition $ω_{f}(u) \neq ω_{f}(v)$ holds; where $ω_{f}(u)=\sum _{x\in N(u)} f(xu)$. Assigning $ω_{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of colors appearing in this proper vertex coloring. The {\it local antimagic chromatic number} of $G$, denoted by $χ_{la}(G)$, is defined as the minimum of $|f|$, wh…
