Date of Award
6-2026
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Ping Zhang, Ph.D.
Second Advisor
Gary Chartrand, Ph.D.
Third Advisor
Jeffrey Strom, Ph.D.
Fourth Advisor
Ebrahim Salehi, Ph.D.
Keywords
Chromatic number, graph coloring, graph theory, total domination, total domination number
Abstract
A question involving a chess piece called a prince on the 8×8 chessboard leads to a concept in graph theory involving total domination. We say a vertex u in a graph G totally dominates a vertex v if u is adjacent to v. A subset S of the vertex set of a graph G is a total dominating set for G if every vertex in G is totally dominated by at least one vertex of S. If S is a total dominating set of G, then σS(v) denotes the number of vertices in S which dominate v. A total dominating set S for a graph G is called a proper total dominating set if σS(u) ̸= σS(v) for every adjacent pair of vertices u and v of G. While every graph without isolated vertices has a total dominating set, not every graph without isolated vertices has a proper total dominating set. We investigate graphs with proper total dominating sets from well known classes of graphs, from the Cartesian product of graphs to trees. Further, for a proper total dominating set S of a graph G, let σS(v) become the color of each vertex v in G. This leads to a proper coloring of G with respect to S which we call a σS coloring of G. The pt-chromatic number χpt(G) of a graph G is the smallest number of distinct colors used by a σS coloring of G among all proper total dominating sets S of G. We present results and open problems from this area of study.
Access Setting
Dissertation-Open Access
Recommended Citation
Osborn, Sawyer Isaac, "From Total Domination to Graph Coloring" (2026). Dissertations. 4275.
https://scholarworks.wmich.edu/dissertations/4275
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