#### Title

Detectable Coloring of Graphs

#### Date of Award

7-2006

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Dr. Ping Zhang

#### Second Advisor

Dr. Gary Chartrand

#### Third Advisor

Dr. Allen Schwenk

#### Fourth Advisor

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 *a**i* is the number of edges incident with *v* that are colored *i* (1 ≤ *i* ≤ *k* ). 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 ≤ *k* ≤ *n* - 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.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Escuadro, Henry E., "Detectable Coloring of Graphs" (2006). *Dissertations*. 939.

http://scholarworks.wmich.edu/dissertations/939

*Abstract*

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## Comments

5th Advisor: Dr. Donald VanderJagt