Related papers: Equitable chromatic threshold of complete multipar…

The cochromatic number $\zeta(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi(G)$ denote the usual chromatic number. Around 1991…

Proportional choosability is a list coloring analogue of equitable coloring. Specifically, a $k$-assignment $L$ for a graph $G$ specifies a list $L(v)$ of $k$ available colors to each $v \in V(G)$. An $L$-coloring assigns a color to each…

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any $c_1,c_2\in C$ with $c_1\ne c_2$ there are in $G$ adjacent vertices $v_1,v_2$ such that $f(v_1)=c_1$ and $f(v_2)=c_2$. The achromatic number of $G$ is the maximum number…

Given a graph $G$, a coloring $c:V(G)\longrightarrow \{1,\ldots,k\}$ such that $c(u)=c(v)=i$ implies that vertices $u$ and $v$ are at distance greater than $i$, is called a packing coloring of $G$. The minimum number of colors in a packing…

We demonstrate that for every positive integer $\Delta$, every K\_4-minor-free graph with maximum degree $\Delta$ admits an equitable coloring with k colors wherek $\ge$ ($\Delta$+3)/2. This bound is tight and confirms a conjecture by Zhang…

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.…

The equitable coloring problem, introduced by Meyer in 1973, has received considerable attention and research. Recently, Wu, Zhang and Li introduced the concept of equitable $(t,k)$-tree-coloring, which can be regarded as a generalization…

A coloring of edges of a graph $G$ is injective if for any two distinct edges $e_1$ and $e_2$, the colors of $e_1$ and $e_2$ are distinct if they are at distance $1$ in $G$ or in a common triangle. Naturally, the injective chromatic index…

An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

A proper edge $t$-coloring of a graph $G$ is a coloring of edges of $G$ with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the same color. The set of colors of edges incident with a vertex $x$ is called…

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest…

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…

An edge-coloured path is monochromatic if all of its edges have the same colour. For a $k$-connected graph $G$, the monochromatic $k$-connection number of $G$, denoted by $mc_k(G)$, is the maximum number of colours in an edge-colouring of…

A graph is perfect if the chromatic number of every induced subgraph equals the size of its largest clique, and an algorithm of Gr\"otschel, Lov\'asz, and Schrijver from 1988 finds an optimal colouring of a perfect graph in polynomial time.…

Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an…

The problem of finding the minimum number of colors to color a graph properly without containing any bicolored copy of a fixed family of subgraphs has been widely studied. Most well-known examples are star coloring and acyclic coloring of…

In this paper, a $k$-edge-coloring of $G$ is any mapping $c:E(G)\longrightarrow [k]$. The edge-coloring $c$ of $G$ naturally defines a vertex-coloring $\sigma_{c}: V(G) \to \mathbb{N}$, where $\sigma_{c}(v)=\sum_{u\in N_G(v)}c(vu)$ for…

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…

The total chromatic number, $\chi''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph,…