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.

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (∆ + 1)vertex coloring and (2∆ − 1)-edge coloring in a graph with maximum degree ∆. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following th...

Clustering algorithms consist in automatically discovering structure of large data set or providing coherent groups of objects independent of any user-defined classes. They are used in many domains: astronomy, information retrieval, image segmentation, biological applications and so on. Several clustering techniques have been proposed [1], there are either hierarchical techniques or partitionin...

An interval edge t − coloring of a graph G is a proper edge coloring of G with colors 1, 2, , t ... such that at least one edge of G is colored by color , 1, 2, , i i t = ... , and the edges incident with each vertex ( ) v V G ∈ are colored by ( ) G d v consecutive colors, where ( ) G d v is the degree of the vertex v in G . In this paper interval edge colorings of bipartite cylinders and bipar...

This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same ...

Let G be an edge-colored graph. A tree T in G is a proper tree if no two adjacent edges of it are assigned the same color. Let k be a fixed integer with 2 ≤ k ≤ n. For a vertex subset S ⊆ V (G) with |S| ≥ 2, a tree is called an S-tree if it connects the vertices of S in G. A k-proper coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G, there exists...