Date of Defense


Date of Graduation




First Advisor

Ping Zhang

Second Advisor

Niloufer Mackey

Third Advisor

John Martino


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.

Access Setting

Honors Thesis-Open Access