**Harmonious Chromatic Number Of A Graph**. The harmonious chromatic number of a graph g, denoted by h (g), is the least number of colors which can be assigned to the vertices of g such that each vertex has exactly one color, adjacent vertices have different colors, and any two edges have different color pairs. Figure 1 shows the central graph of q 3 with harmonious coloring.

05c15 key words and phrases. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Zhikang lu, department of mathematics hangzou teacher's college hangzou, zhejiang, 310036 peoples republic of china.

## A harmonious colouring of a simple graph g is a proper vertex colouring such that each pair of colours appears together on at most one edge.

The achromatic number of a graph is the. The harmonious chromatic number of a graph is the least number of colours in a vertex colouring such that each pair of colours appears on at most one edge. The chromatic polynomial counts the number of ways a graph can be colored using some of a given number of colors. Prism graph by mansuri et al.

The harmonious chromatic number is the minimal number of hues in such a coloring. The minimum number of colors required to have a valid harmonious coloring, that is to ﬁnd the harmonious chromatic number of a graph, as deﬁned next. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. The harmonious chromatic number of a graph is the least number of colours in such a colouring, where g is a finite un directed graph with no loops and multiple edges.

### The achromatic number of a graph is the.

This parameter is based on the idea of coloring the vertices of a graph with integers (modulo k) in such a way that the colors on adjacent vertices are as far apart as possible. Each graph has a harmonious coloring, since it does the trick to allocate each vertex a different color. The harmonious chromatic number h(g) is the least number of colours in such a colouring. On an upper bound for the harmonious chromatic number of a graph.

### The complexity of the corresponding undirected case is not known.

This chapter describes the harmonious chromatic number of a graph. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. 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. We also look at families of graphs where the harmonious coloring is minimal.

### We show that there is a natural number n (d) such that if t is any tree with m ≥ n (d) edges and maximum degree at most d, then.

Note that the problem of determining the harmonious chromatic number of connected cographs is trivial, since in such a graph each vertex must receive a distinct color as it is at distance at most 2 from all other vertices [6. A harmonious colouring of a simple graph g is a proper vertex colouring such that each pair of colours appears together on at most one edge. Here a color pair for edge a is the set of colors on the vertices of e. For example, using three colors, the graph in the adjacent image can be colored in 12 ways.

The harmonious chromatic number is the minimal number of hues in such a coloring. Harmonious chromatic number of central graph of quadrilateral snakes 3 coloring, therefore it is minimum. Paths and cycles are among the rst graphs whose harmonious chromatic numbers have been established 3. 09/02/2020 ∙ by florent foucaud, et al.