Related papers: Coloring of graphs associated to zero-divisors

A graph $G$ is said to be equitably $c$-colorable if its vertices can be partitioned into $c$ independent sets that pairwise differ in size by at most one. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree…

Recall that the minimum number of colors that allow a proper coloring of graph $G$ is called the chromatic number of $G$ and denoted by $\chi(G).$ In this paper the concepts of $\chi$'-chromatic sum and $\chi^+$-chromatic sum are…

An $n$-subdivision of a graph $G$ is a graph constructed by replacing a path of length $n$ instead of each edge of $G$ and an $m$-power of $G$ is a graph with the same vertices as $G$ and any two vertices of $G$ at distance at most $m$ are…

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…

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…

For a multigraph $G$, $\chi'(G)$ denotes the chromatic index of $G$, $\Delta(G)$ the maximum degree of $G$, and $\Gamma(G) = \max\left\{\left\lceil \frac{2|E(H)|}{|V(H)|-1} \right\rceil: H \subseteq G \text{ and } |V(H)| \text{…

A class of graphs $\cal G$ is said to be \emph{near optimal colorable} if there exists a constant $c\in \mathbb{N}$ such that every graph $G\in \cal G$ satisfies $\chi(G) \leq \max\{c, \omega(G)\}$, where $\chi(G)$ and $\omega(G)$…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

Let $G$ be a graph with chromatic number $\chi$, maximum degree $\Delta$ and clique number $\omega$. Reed's conjecture states that $\chi \leq \lceil (1-\varepsilon)(\Delta + 1) + \varepsilon\omega \rceil$ for all $\varepsilon \leq 1/2$. It…

Let $G$ be a connected graph with maximum degree $\Delta \ge 3$. We investigate the upper bound for the chromatic number $\chi_\gamma(G)$ of the power graph $G^\gamma$. It was proved that $\chi_\gamma(G)…

Given a graph $G$ and a natural number $k$, the $k$-recolouring graph $\mathcal{C}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and whose edges link pairs of colourings which differ at exactly one vertex of $G$. Recently,…

Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with a same color, if there is a shortest $x,y$-path whose internal vertices have different colors than $x,y$. The smallest…

Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq…

For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$ and $E(G)$ the sets of its vertices and edges, respectively. A function $\varphi:E(G)\rightarrow \{1,...,t\}$ is called a proper edge $t$-coloring of a graph $G$,…

The chromatic number $\chi((G,\sigma))$ of a signed graph $(G,\sigma)$ is the smallest number $k$ for which there is a function $c : V(G) \rightarrow \mathbb{Z}_k$ such that $c(v) \not= \sigma(e) c(w)$ for every edge $e = vw$. Let…

A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex…

Let $G$ and $H$ be two vertex disjoint graphs. The {\em union} $G\cup H$ is the graph with $V(G\cup H)=V(G)\cup V(H)$ and $E(G\cup H)=E(G)\cup E(H)$. We use $P_k$ to denote a {\em path} on $k$ vertices, use {\em house} to denote the…

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $\Delta \ge 9$ satisfies $\chi \le \max\{\omega, \Delta-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main…

A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number $\chi_s(G)$ is the smallest number of colors required to obtain a star…

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the…