Harmonious Coloring In Graph Theory

The harmonious chromatic number of g denoted by χ h g is the least number of colors in a harmonious coloring of g. Abstract in this report we define a new coloring of graphs namely harmonious coloring of graphs which arises as an extension of harmonious and graceful numbering of graphs.

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The hermonious coloring number of the graph g hc g is the smallest number of colors needed to label the vertices of g such that adjacent vertices received different colors and no two edges are incident with the same color pair.

Harmonious coloring in graph theory. Graph theory coloring. A harmonious coloring of a simple graph is the proper vertex coloring such that each pair of colors appears together on at most one edge. K for which g admits a harmonious coloring with k colors.

This is called a vertex coloring. It is an assignment of labels traditionally called colors to elements of a graph subject to certain constraints. In graph theory graph coloring is a special case of graph labeling.

Given a simple graph g a harmonious coloring of g is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious coloring problem of a graph is to find the minimum number of colors needed to color the vertices of a graph such that the color pairs of end vertices of every edge are distinct. The harmonious chromatic number χ h g of a graph g is the minimum number of colors needed for any harmonious coloring of g.

The harmonious coloring problem of a graph is to ﬁnd the minimum number of colors needed to color the vertices of a graph such that the color pairs of end vertices of every edge are distinct. Graph coloring is nothing but a simple way of labelling graph components such as vertices edges and regions under some constraints. We show that the.

The harmonious coloring problem of a graph is to ﬁnd the minimum number of colors needed to color the vertices of a graph such that the color pairs of end vertices of every edge are distinct. The paper gives the structural properties and the. In its simplest form it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color.

In graph theory a harmonious coloring is a proper vertex coloring in which every pair of colors appears on at most one pair of adjacent vertices. Hopcroft and krishnamoorthy first established the abstraction of the harmonious coloring problem and proved the np completeness for general graphs 4. In a graph no two adjacent vertices adjacent edges or adjacent regions are colored with minimum number of colors.

The harmonious chromatic number is the least integer.

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