# AN INVESTIGATION OF ACHROMATIC AND HARMONIOUS CHROMATIC NUMBER OF CERTAIN CLASSES OF GRAPHS

2024

Thesis

## Degree Name

Master of Science in Mathematics

## Department

Mathematics and Applied Mathematical Sciences

Nancy Eaton

## Abstract

One major area of graph theory is the area of graph coloring. In this thesis, we specifically focus on problems in graph coloring that relate to proper vertex colorings. Proper vertex colorings are ones in which no two adjacent vertices are given the same color. Because of this, we are able to sort our vertices into independent groups based upon the colors they are assigned. Proper vertex colorings have been studied by mathematicians because of their vast applications to the real world. These applications include scheduling problems, solving Sudoku, allocating jobs/assignments to workers, seating plans and more.

In this thesis, we focus on two types of graph coloring problems: the achromatic number and the harmonious chromatic number. We specifically focus on graphs whose chromatic number is either 2 or 3, which include paths, cycles, bipartite graphs, and trees. Given a graph, G, a harmonious coloring of G is a proper vertex coloring such that each color pair is used at most once. The minimum number of colors used to create such a coloring is known as the harmonious chromatic number of G, denoted χH (G). A complete coloring of a graph G is a proper vertex coloring such that each color pair is used at least once. The maximum number of colors used to create such a coloring is known as the achromatic number of G, denoted ψ(G).

We note that it is always true that ψ(G) ≤ χH (G) for any graph G. Here we provide bounds for these numbers, existence of graphs where they are the same, and instances where the distance between the two numbers is maximized.

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