To prove that the chromatic number of a graph $G$ will not exceed the maximum degree of its vertices by more than one, we can make use of Greedy Coloring and the concept of the graph's maximum degree.

Definitions:

1. Chromatic Number ($\chi(G)$): The smallest number of colors required to color the vertices of a graph $G$ such that no two adjacent vertices share the same color.
2. Maximum Degree ($\Delta(G)$): The highest degree (i.e., number of edges) among all vertices in the graph $G$.

Theorem to Prove:

The chromatic number $\chi(G)$ of any graph $G$ is bounded by: $\chi(G) \leq \Delta(G) + 1$

Proof:

The proof follows from the Greedy Coloring Algorithm.

Step-by-Step Proof:

1. Ordering the Vertices: Consider an arbitrary ordering of the vertices of the graph $G$,