The inflation $G_{I}$ of a graph $G$ with $n(G)$ vertices and $m(G)$ edges is obtained from $G$ by replacing every vertex of degree $d$ of $G$ by a clique, which is isomorph to the complete graph $K_{d}$, and each edge $(x_{i},x_{j})$ of $G$ is replaced by an edge $(u,v)$ in such a way that $uin X_{i}$, $vin X_{j}$, and two different edges of $G$ are replaced by non-adjacent edges of $G_{I}$. T...

The total dominator coloring of a graph is the such that each object (vertex or edge) adjacent incident to every some color class. minimum number classes called chromatic graph. In (A.P. Kazemi, F. Kazemnejad and S. Moradi, Contrib. Discrete Math. (2022).), authors initiated study found useful results, presented problems. Finding numbers cycles paths were two them which we consider here.

Let R be a commutative ring with $Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $T(Gamma(R))$. It is the (undirected) graph with all elements of R as vertices, and for distinct $x, yin R$, the vertices $x$ and $y$ are adjacent if and only if $x + yinZ(R)$. We study the chromatic number and edge connectivity of this graph.

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G∗ be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G′ has game chromatic number at most k(k+m − 1). As a consequence, the Cartesian product of two forests has game c...

In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph G as a mapping φ : V (G) → Z such that for any two adjacent vertices u and v the colour φ(u) is different from the colour σ(uv)φ(v), where is σ(uv) is the sign of the edge uv. The substantial part of Zaslavsky’s research concentrated on polynomial invariants related to signed graph colourings rather than on...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D(~ G), the maximum number of descendants of a gi...

We consider vertex coloring of an acyclic digraph ~ G in such a way that two vertices which have a common ancestor in ~ G receive distinct colors. Such colorings arise in a natural way when bounding space for various genetic data for efficient analysis. We discuss the corresponding down-chromatic number and derive an upper bound as a function of D(~ G), the maximum number of descendants of a gi...