Chromatic Theory of Graphs

Abstract

Throughout this section Г will denote the graph (V,E). A vertex m-coloring of Г is a surjection h of V onto an m-set C, subject to the condition: If x₁, x₂ ∈P₂ (V) and h(x₁) = h(x₂) , then x₁ and x₂ are not incident. The elements of C are called colors and the sets h ⁻¹ j for j ∈ C are called color classes. If x ∈ V and h(x) = j, one also says, “x has been colored j.” We say that Г is vertex m-colorable if Г admits a vertex m-coloring. The vertex chromatic number of Г is

$${\chi _o}\left( \Gamma \right) = \min \left\{ m \right.:\Gamma \left. {has a vertex m - coloring} \right\}$$

.