Abstract
This paper studies various results on vertex colorings of simple connected graphs, chromatic number, chromatic polynomials and some Algebraic properties of chromatic polynomials. Results were obtained on the roots of chromatic polynomials of simple connected graphs based on Read’s conjecture. The chromatic number of every graph is the minimum number of colors to properly color the graph. Chromatic polynomial of a graph is a polynomial in integer and the leading coefficient of chromatic polynomial of a graph of order n and size <i>m</i> is always 1, whose coefficient alternate in sign. Through the application of famous graph theorem (the hand shaking lemma) by whiskey which states that: “the order of a graph twice its size”. Hence, every graph has a chromatic polynomial but not all polynomials are chromatic. For example, the polynomial λ<sup>5</sup> − 11 λ<sup>4</sup> + 14 λ<sup>3</sup> − 6 λ<sup>2</sup> + 2 λ is a polynomial for a graph on five vertices and eleven edges which does not exists. Because the maximum number size for a graph of order five is ten. The paper equally gave some practical applications of Vertex coloring in real life situations such as scheduling, allocation of channels to television and radio stations, separation of chemicals and traffic light signals.
Highlights
A graph G is a finite nonempty set V of objects called vertices together with a set E of 2-element subsets of V called edges
Graphs are typically represented by diagrams in which each vertex is represented by a point or small circle and each edge is represented by a line segment or curve joining the corresponding small circles
A proper vertex coloring of a graph G is a vertex coloring such that the end points of each edge are assigned two different colors [8]
Summary
A graph G is a finite nonempty set V of objects called vertices (the singular is vertex) together with a set E of 2-element subsets of V called edges. The number of vertices in a graph G is the order of G and the number of edges is the size of G. Graphs are typically represented by diagrams in which each vertex is represented by a point or small circle (open or solid) and each edge is represented by a line segment or curve joining the corresponding small circles. The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved. For his accomplishment Kempe was elected a Fellow of the Royal Society and later President of the London Mathematical Society. Heawood himself modified that thought [10]
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