Candidate: Henry E. Escuadro

Degree of: Doctor of Philosophy

Department: Mathematics

Title: Detectable Colorings of Graphs

Committee:
Dr. Ping Zhang, Chair
Dr. Gary Chartrand
Dr. Allen Schwenk
Dr. Clifton Ealy
Dr. Donald VanderJagt

Date: Thursday, January 26, 2006 3:00 p.m.- 5:00 p.m.
Alavi Commons, 6625 Everett Tower

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 problem of 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 3 or more and let c: E(G) → {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 is the k – tuple code (v) = (a1, a2,…, ak), where ai is the number of edges incident with v that are colored i (1≤ i ≤ k).  The coloring c is called detectable if distinct vertices have distinct color codes; while the detection number det(G) of G is the minimum positive integer k for which G has a detectable k – coloring.

We determine the detection numbers of several well-known classes of graphs, study the properties of detectable colorings of regular graphs, and establish bounds for the detection number of a graph in terms of graphical parameters.  We also investigate a number of extremal problems dealing with detectable colorings.  Since (over) detectable colorings can be looked at terms of factorizations, we study the related concept of detectable factorizations of graphs.

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