In this paper some new results on the acyclic-edge coloring , f -edge coloring, g-edge cover coloring, (g, f )-coloring and equitable edge-coloring of graphs are introduced. In particular, some new results related to the above colorings obtained by us are given. Some new problems and conjectures are presented.

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

In this paper we study a generalization of both proper edge-coloring and strong edge-coloring: k-intersection edge-coloring, introduced by Muthu, Narayanan and Subramanian [18]. In this coloring, the set S(v) of colors used by edges incident to a vertex v does not intersect S(u) on more than k colors when u and v are adjacent. We provide some sharp upper and lower bounds for χk-int for several ...

Reconfiguration problems arise when we wish to find a step-by-step transformation between two feasible solutions of a problem such that all intermediate results are also feasible. Recently, Ito et al. [5] proposed a framework of reconfiguration problems, and gave complexity and approximability results for reconfiguration problems derived from several well-known problems, such as independent set...

We consider the problem of list edge coloring for planar graphs. Edge coloring is the problem of coloring the edges while ensuring that two edges that are incident receive different colors. A graph is k-edge-choosable if for any assignment of k colors to every edge, there is an edge coloring such that the color of every edge belongs to its color assignment. Vizing conjectured in 1965 that every...

A graph edge is d-coloring redundant if the removal of the edge does not change the set of dcolorings of the graph. Graphs that are too sparse or too dense do not have coloring redundant edges. Tight upper and lower bounds on the number of edges in a graph in order for the graph to have a coloring redundant edge are proven. Two constructions link the class of graphs with a coloring redundant ed...

An acyclic edge coloring of a graph G is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index χa(G) of a graph G is the least number of colors in an acyclic edge coloring of G. It was conjectured that χa(G) ≤ ∆(G) + 2 for any simple graph G with maximum degree ∆(G). In this paper, we prove that every planar graph G admits an acyclic edg...

We propose the notion of a majority $k$-edge-coloring graph $G$, which is an edge-coloring $G$ with $k$ colors such that, for every vertex $u$ at most half edges incident have same color. show best possible results that minimum degree least $2$ has $4$-edge-coloring, and $4$ $3$-edge-coloring. Furthermore, we discuss natural variation edge-colorings some related open problems.