#### Date of Defense

Fall 11-18-2005

#### Department

Mathematics

#### First Advisor

Ping Zhang, Mathematics

#### Second Advisor

Arthur White, Mathematics

#### Abstract

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) = (*a*_{0}, *a*_{1},...,*a _{k}*) where

*a*

_{0}is the color assigned to

*v*and for 1 ≤

*i*≤

*k*,

*a*is the number of vertices adjacent to

_{i}*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 ≤

*k*≤

*n*, 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.

#### Recommended Citation

Radcliffe, Mary Lynn, "On the Irregular Colorings of Graphs" (2005). *Honors Theses*. 294.

https://scholarworks.wmich.edu/honors_theses/294

#### Access Setting

Honors Thesis-Campus Only