Tight Bounds on 1-Harmonious Coloring of Certain Graphs

Abstract

Graph coloring is one of the most studied problems in graph theory due to its important applications in task scheduling and pattern recognition. The main aim of the problem is to assign colors to the elements of a graph such as vertices and/or edges subject to certain constraints. The 1-harmonious coloring is a kind of vertex coloring such that the color pairs of end vertices of every edge are different only for adjacent edges and the optimal constraint that the least number of colors is to be used. In this paper, we investigate the graphs in which we attain the sharp bound on 1-harmonious coloring. Our investigation consists of a collection of basic graphs like a complete graph, wheel, star, tree, fan, and interconnection networks such as a mesh-derived network, generalized honeycomb network, complete multipartite graph, butterfly, and Benes networks. We also give a systematic and elegant way of coloring for these structures.

Dom-Chromatic Number o f Wrapped Butterfly & Bloom Graphs

“…The study on domination and coloring in wrapped butterfly and bloom graphs are interesting areas of research in graph theory. In wrapped butterfly network, few studies like 1-harmonious coloring (1) , T-coloring (2) , ST-coloring (2) , total domination (3) and domination of its digraph (4) were carried on. In the above research work, only one concept, either coloring or domination were applied, whereas in this paper both the concepts are applied together to determine the dom-chromatic number of wrapped butterfly graph.…”

1-Harmonious coloring of triangular snakes

“…The minimum number of colors are required for this coloring is called its chromatic number, and is generally denoted by χ(G). The 1-harmonious coloring [4] is a kind of vertex coloring such that the color pairs of end vertices of every edge are different only for adjacent edges and a minimum number of colors are required for this coloring is called the 1-harmonious chromatic number, denoted by h 1 (G). A triangular snake [2, 5-7, 9, 10] is a triangular cactus whose blockcutpointgraph is a path (a triangular snake is obtained from a path u 1 , u 2 , ..., u n by joining u i and u i+1 to a new vertex w i for i = 1, 2, ..., n − 1).…”