## Dissertations

#### Title

Detectable Coloring of Graphs

7-2006

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Ping Zhang

Dr. Gary Chartrand

Dr. Allen Schwenk

Dr. Clifton Ealy

#### Abstract

A basic problem in graph theory is to distinguish the vertices of a connected graph from one another in some manner. In this study, we investigate the problemof coloring the edges of a graph in a manner that distinguishes the vertices of the graph. The method we use combines many of the features of previously introduced methods.

Let G be a connected graph of order n ≥ 3 and let c : E (G ) [arrow right] {1,2,...,k } be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G , the color code of v with respect to c is the k -tuple c ( v ) = (α1 ,α2 ,···,α k ), where ai is the number of edges incident with v that are colored i (1 ≤ ik ). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det( G ) of G is the minimum positive integer k for which G has a detectable k -coloring.

The detection number of stars, double stars, cycles, paths, complete graphs, and complete bipartite graphs are determined. It is also shown that a pair k , n ofpositive integers is realizable as the detection number and the order of some nontrivial connected graph if and only if k = n = 3 or 2 ≤ kn - 1.

Extremal problems on detectable colorings of graphs are investigated in this study. If G is a connected graph of order n and size m , then the number of edges that must be deleted from G to obtain a spanning tree of G is m - n + 1. The number m - n + 1 is called the cycle rank of G . For integers [Special characters omitted.] and n , where [Special characters omitted.] ≥ 0 and n ≥ [Special characters omitted.] , let [Special characters omitted.] (n ) denote the maximum detection number among all connected graphs of order n with cycle rank [Special characters omitted.] and let [Special characters omitted.] (n ) denote the minimum detection number among all connected graphs of order n with cycle rank [Special characters omitted.] . Hence, if [Special characters omitted.] denotes the set of all connected graphs of order n with cycle rank [Special characters omitted.] , then[Special characters omitted.]

*Full abstract attached as separate file.