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.
Highlights
In mathematical sciences, graph coloring problems are very easy to state and visualize, but they have many aspects that are exceptionally difficult to solve [1]
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
It was seen that this is a problem in graph theory, whether it is always possible to color the regions of every planar graph so that every two adjacent regions are colored differently [2]
Summary
Graph coloring problems are very easy to state and visualize, but they have many aspects that are exceptionally difficult to solve [1]. It was seen that this is a problem in graph theory, whether it is always possible to color the regions of every planar graph (embedded in the plane) so that every two adjacent regions are colored differently [2] It took more than 160 years and collective efforts to prove the simple-sounding proposition that four colors are sufficient to color the vertices of a planar graph properly [3].
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