Related papers: Circular Game Coloring of Signed Graphs

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our…

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by…

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are…

Let $G$ be a graph whose each component has order at least 3. Let $s : E(G) \rightarrow \mathbb{Z}_k$ for some integer $k\geq 2$ be an improper edge coloring of $G$ (where adjacent edges may be assigned the same color). If the induced…

Given a graph $G$ and a list assignment $L$ for $G$, the indicated $L$-colouring game on $G$ is played by two players: Ann and Ben. In each round, Ann chooses an uncoloured vertex $v$, and Ben colours $v$ with a colour from $L(v)$ that is…

Investigating the equality of the chromatic number and the circular chromatic number of graphs has been an active stream of research for last decades. In this regard, Habolhassan and Zhu [Circular chromatic number of Kneser graphs, Journal…

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…

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $\chi'(G \setminus F)<\chi'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{\chi'}(G)$, and such a (smallest) set $F$ is…

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…

Game coloring is a well-studied two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. An `eternal' version of game coloring is introduced in this paper in which the vertices…

Assume $G$ is a graph. We view $G$ as a symmetric digraph, in which each edge $uv$ of $G$ is replaced by a pair of opposite arcs $e=(u,v)$ and $e^{-1}=(v,u)$. Assume $S$ is an inverse closed subset of permutations of positive integers. We…

An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring}…

A \emph{coloring} of a graph $G$ is a map $f:V(G)\to \mathbb{Z}^+$ such that $f(v)\ne f(w)$ for all $vw\in E(G)$. A coloring $f$ is an \emph{odd-sum} coloring if $\sum_{w\in N[v]}f(w)$ is odd, for each vertex $v\in V(G)$. The \emph{odd-sum…

Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that…

A signed graph is said to be sign-symmetric if it is switching isomorphic to its negation. Bipartite signed graphs are trivially sign-symmetric. We give new constructions of non-bipartite sign-symmetric signed graphs. Sign-symmetric signed…

An injective edge-coloring $c$ of a graph $G$ is an edge-coloring such that if $e_1$, $e_2$, and $e_3$ are three consecutive edges in $G$ (they are consecutive if they form a path or a cycle of length three), then $e_1$ and $e_3$ receive…

We prove that for every $t\in \mathbb{N}$ there is a constant $\gamma_t$ such that every graph with twin-width at most $t$ and clique number $\omega$ has chromatic number bounded by $2^{\gamma_t \log^{4t+3} \omega}$. In other words, we…

We define $G$-cospectrality of two $G$-gain graphs $(\Gamma,\psi)$ and $(\Gamma',\psi')$, proving that it is a switching isomorphism invariant. When $G$ is a finite group, we prove that $G$-cospectrality is equivalent to cospectrality with…

An $r$-dynamic $k$-coloring of a graph $G$ is a proper vertex $k$-coloring such that the neighbors of any vertex $v$ receive at least $\min\{r,{\rm deg}(v)\}$ different colors. The $r$-dynamic chromatic number of $G$, $\chi_r(G)$, is…

Let $G$ be a graph, $\chi(G)$ be the minimal number of colors which can be assigned to the vertices of $G$ in such a way that every two adjacent vertices have different colors and $\omega(G)$ to be the least upper bound of the size of the…