An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov and Zaks (and earlier by Fiamcik) that a′(G)≤ +2, where = (G) denotes the maximum degree of the graph. Alon e...

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.

A coloring of a graph G is an assignment of colors to the vertices of G such that any two vertices receive distinct colors whenever they are adjacent. An acyclic coloring of G is a coloring such that no cycle of G receives exactly two colors, and the acyclic chromatic number χA(G) of a graph G is the minimum number of colors in any such coloring of G. Given a graph G and an integer k, determini...

A theta graph Θ2,1,2 is a obtained by joining two vertices three internally disjoint paths of lengths 2, 1, and 2. neighbor sum distinguishing (NSD) total coloring ϕ G proper such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set edges incident with vertex u. In 2015, Pilśniak Woźniak introduced this conjectured every maximum degree Δ admits an NSD (Δ+3)-...

In this paper, we give a relatively simple though very efficient way to color the d-dimensional grid G(n1, n2, . . . , nd) (with ni vertices in each dimension 1 i d), for two different types of vertex colorings: (1) acyclic coloring of graphs, in which we color the vertices such that (i) no two neighbors are assigned the same color and (ii) for any two colors i and j , the subgraph induced by t...

An edge irregular total k-labeling of a graph G is a labeling of the vertices and edges with labels 1, 2, . . . , k such that the weights of any two different edges are distinct, where the weight of an edge is the sum of the label of the edge itself and the labels of its two end vertices. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregul...

A graph G is called k-ordered if for every sequence of k distinct vertices there is a cycle traversing these vertices in the given order. In the present paper we consider two novel generalizations of this concept, k-vertex-edge-ordered and strongly k-vertex-edge-ordered . We prove the following results for a chordal graph G: (a) G is (2k − 3)-connected if and only if it is k-vertex-edge-ordered...

Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...

In an ordinary edge-coloring of a graph each color appears at each vertex at most once. An f -coloring is a generalized edge-coloring in which each color appears at each vertex v at most f(v) times where f(v) is a positive integer assigned to v. This paper gives efficient sequential and parallel algorithms to find ordinary edge-colorings and f -colorings for various classes of graphs such as bi...