Date of Defense

Fall 11-18-2005



First Advisor

Ping Zhang, Mathematics

Second Advisor

Arthur White, Mathematics


Let G be a connected graph and let c : V(G) → {1,2,...,k} be a proper coloring of the vertices of G for some positive integer k. The color code of a vertex v of G (with respect to c) is the ordered (k + 1)-tuple code(v) = (a0, a1,...,ak) where a0 is the color assigned to v and for 1 ≤ ik, ai is the number of vertices adjacent to v that are colored i. The coloring c is irregular if distinct vertices have distinct color codes and the irregular chromatic number Xir(G) of G is the minimum positive integer k for which G has an irregular k-coloring. Characterizations of connected graphs of order n having irregular chromatic numbers 2 or n are established. For a pair k, n of integers with 2 ≤ kn, it is shown that there exists a connected graph of order n having irregular chromatic number k if and only if (k,n) ≠ (2,n) for some odd integer n. Irregular chromatic numbers of cycles are investigated. The author also studies the irregular chromatic numbers of disconnected graphs and Nordhaus-Gaddum inequalities for the irregular chromatic number of a graph.

Access Setting

Honors Thesis-Campus Only