#### Date of Defense

4-19-2017

#### Date of Graduation

4-2017

#### Department

Mathematics

#### First Advisor

Ping Zhang

#### Second Advisor

Niloufer Mackey

#### Third Advisor

John Martino

#### Abstract

There have been numerous studies using a variety of methods for the purpose of uniquely distinguishing every two adjacent vertices of a graph. Many of these methods have involved graph colorings. The most studied colorings are proper colorings. A proper coloring of a graph *G* is an assignment of colors to the vertices of *G* such that adjacent vertices are assigned distinct colors. The minimum number of colors required in a proper coloring of *G* is the chromatic number of *G*. In our work, we introduce a new coloring that induces a (nearly) proper coloring. Two vertices *u* and* v* in a nontrivial connected graph *G* are twins if u and v have the same neighbors in V (G) - {u,v}. If u and v are adjacent, they are referred to as true twins; while if *u* and *v* are nonadjacent, they are false twins. For a positive integer *k*, let *c* be a coloring of a graph G using colors in the set Nk = {1,2,...,k}. Define another coloring *s* of *G* such that the color *s(v) *of a vertex *v* is the sum of the colors of all vertices in the closed neighborhood of *v.* Then *c* is called a closed sigma* k*-coloring if *s(u)* ≠ *s(v)* for all pairs* u*, *v* of adjacent vertices that are not true twins. The minimum *k* for which *G* has a closed sigma *k*-coloring is the closed sigma chromatic number of *G*, denoted by *Xs(G). *We study closed sigma colorings of graphs and the relationship among closed sigma colorings and other graphical parameters. Closed sigma chromatic numbers have been determined for several well-known classes of connected graphs. Other results and open questions are presented.

#### Recommended Citation

Hallas, James, "Sum-Defined Colorings in Graphs" (2017). *Honors Theses*. 2816.

https://scholarworks.wmich.edu/honors_theses/2816

#### Access Setting

Honors Thesis-Open Access