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For a proper vertex coloring $c$ of a graph $G$, let $\varphi_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $\chi(H)$ and the number of colors used by $c$ to color $H$. We…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

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…

It is known that the inequality $$ \frac{\chi(G)(\chi(G)-1)}{2} + |V| - \chi(G) \leq |E|$$ holds for all connected graphs, where $\chi(G)$ denotes the chromatic number of $G$. We prove that equality holds whenever the graph consists of a…

It is known (Bollob\'{a}s (1978); Kostochka and Mazurova (1977)) that there exist graphs of maximum degree $\Delta$ and of arbitrarily large girth whose chromatic number is at least $c \Delta / \log \Delta$. We show an analogous result for…

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…

A proper edge coloring of a graph without any bichromatic cycles is said to be an acyclic edge coloring of the graph. The acyclic chromatic index of a graph $G$ denoted by $a'(G)$, is the minimum integer $k$ such that $G$ has an acyclic…

By a graph we mean a finite undirected graph having multiple edges but no loops. Given a graph property $\mathcal{P}$, a $\mathcal{P}$-coloring of a graph $G$ with color set $C$ is a mapping $\f:V(G)\to C$ such that for each color $c\in C$…

Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal…

We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $G$ is a proper edge coloring of $G$ such that each color class is dominated by at least one edge of $G$. The minimum number of colors among…

A graph is called odd (respectively, even) if every vertex has odd (respectively, even) degree. Gallai proved that every graph can be partitioned into two even induced subgraphs, or into an odd and an even induced subgraph. We refer to a…

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…

The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been…

In this paper, we have obtained the total chromatic number of some classes of Cayley graphs, odd graphs and mock threshold graphs.

We introduce joins of signed graphs and explore the chromatic number of the all-positive and all-negative joins. We prove an analogue to the theorem that the chromatic number of the join of two graphs equals the sum of their chromatic…

Motivated by the definition of linear coloring on simplicial complexes, recently introduced in the context of algebraic topology \cite{Civan}, and the framework through which it was studied, we introduce the linear coloring on graphs. We…

A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G) \rightarrow \{+1,-1\}$. An incidence of $G$ is a pair $(v,e)$, where $v$ is one of the end vertices of an edge $e \in E(G)$. A proper $q$-edge coloring…

A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo…

The dichromatic number and the diachromatic number are generalizations of the chromatic number and the achromatic number for digraphs considering acyclic colorings. In this paper, we determine the diachromatic number of digraphs arising…

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $\chi_S(G)$. In this paper, $\chi_S$-critical graphs are introduced as the graphs $G$ such that…