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...

In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erd˝ os and, independently, Albertson et al., proved that every graph of maximum degree at most 3 has a 2-subcoloring. We point out that this fact is best possible with respect to degree constr...

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-coloring with small or optimal λ arises in the context of radio frequency assignment. We show that the problem of finding the...

In recent years, various heuristic optimization methods have been developed. Many of these methods are inspired by swarm behaviors in nature, such as particle swarm optimization (PSO), firefly algorithm (FA) and cuckoo optimization algorithm (COA). Recently introduced COA, has proven its excellent capabilities, such as faster convergence and better global minimum achievement. In this paper a ne...

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...

On Sum Coloring of Graphs Mohammadreza Salavatipour Master of Science Graduate Department of Computer Science University of Toronto 2000 The sum coloring problem asks to nd a vertex coloring of a given graph G, using natural numbers, such that the total sum of the colors of vertices is minimized amongst all proper vertex colorings of G. This minimum total sum is the chromatic sum of the graph, ...

This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing’s theorem states that the edge coloring of a simple graph G requires either ∆ or ∆ + 1 colors, where ∆ is the maximum vertex degree of G. Holyer proved that it is NP-complete to decide whether G is ∆-edge-colorable even for cubic graphs. By introducing the concept of co...