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 $g$ denoted by $chi_a '(g)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. the maximum degree in $g$ denoted by $delta(g)$, is the lower bound for $chi_a '(g)$. $p$-cuts introduced in this paper acts as a powerfu...

A fuzzy graph is a symmetric binary fuzzy relation on a fuzzy subset. The concept of fuzzy sets and fuzzy relations was introduced by L.A.Zadeh in 1965cite{zl} and further studiedcite{ka}. It was Rosenfeldcite{ra} who considered fuzzy relations on fuzzy sets and developed the theory of fuzzy graphs in 1975. The concepts of fuzzy trees, blocks, bridges and cut nodes in fuzzy graph has been studi...

Coloring games are combinatorial games where the players alternate painting uncolored vertices of a graph one of k > 0 colors. Each different ruleset specifies that game’s coloring constraints. This paper investigates six impartial rulesets (five new), derived from previously-studied graph coloring schemes, including proper map coloring, oriented coloring, 2-distance coloring, weak coloring, an...

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 $G$ denoted by $chi_a '(G)$ is the minimum number $k$ such that there is an acyclic edge coloring using $k$ colors. The maximum degree in $G$ denoted by $Delta(G)$, is the lower bound for $chi_a '(G)$. $P$-cuts introduced in this paper acts as a powerfu...

A new variation of the coloring problem, μ-coloring, is defined in this paper. Given a graph G and a function μ, a μ-coloring is a coloring where each vertex v of G must receive a color at most μ(v). It is proved that μ-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. The notion of perfection is extended for μ-coloring, l...

Dvo?ák and Postle introduced the concept of DP-coloring to overcome some difficulties in list coloring. Sittitrai Nakprasit combined defective coloring define a new coloring—relaxed DP-coloring. For relaxed DP-coloring, Sribunhung et al. proved that planar graphs without 4- 7-cycles are DP-(0, 2, 2)-colorable. Li 4, 8-cycles or 9-cycles DP-(1, 1, 1)-colorable. Lu Zhu 5-cycles, 6-cycles, In this...

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, such that no two adjacent or incident elements receive the same color. Let L(x) be a set of colors assigned to each element x of G. Then a list total coloring of G is a total coloring such that each element x receives a color contained in L(x). The list total coloring problem asks whether G has a list tot...