Date of Award

6-2018

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. Ebrahim Salehi

Keywords

Graph theory, vertex colorings, edge colorings, majestic coloring, majestic t-tone coloring

Abstract

An edge coloring of a nonempty graph G is an assignment of colors to the edges of G. In an unrestricted edge coloring, adjacent edges of G may be colored the same. If every two adjacent edges of G are colored differently, then this edge coloring is proper and the minimum number of colors in a proper edge coloring of G is the chromatic index χ/(G) of G. A proper vertex coloring of a nontrivial graph G is an assignment of colors to the vertices of G such that every two adjacent vertices of G are colored differently. The minimum number of colors in a proper vertex coloring of G is the chromatic number χ(G) of G. In this work, we study a proper vertex coloring of a graph that is induced by an unrestricted edge coloring of the graph.

For a connected graph G of order at least 3, let c : E(G) → {1, 2, . . . , k} be an

unrestricted edge coloring of G where adjacent edges may be colored the same. Then c induces a vertex coloring c/ of G obtained by assigning to each vertex v of G the set of colors of the edges incident with v. The edge coloring c is called a majestic k-edge coloring of G if the induced vertex coloring c/ is a proper vertex coloring of G. The minimum positive integer k for which a graph G has a majestic k-edge coloring is the majestic chromatic index of G and and denoted by maj(G). For a graph G with majestic chromatic index k, the minimum number of distinct vertex colors induced by a majestic k-edge coloring is the majestic chromatic number of G and denoted by ψ(G). Thus, ψ(G) is at least as large as the chromatic number χ(G) of a graph G.

Majestic chromatic indexes and numbers are determined for several well-known classes of graphs, including complete graphs, complete multipartite graphs, paths, cycles and connected bipartite graphs and the Cartesian products G □ K2 when G is a bipartite graph as well as two classes of nonbipartite graphs G, namely odd cycles and complete multipartite graph. Sharp bounds have been established for these two parameters and relationship between the majestic chromatic number and the chromatic number of a graph has been studied.

The idea of assigning a color a to an edge of a graph can be looking as assigning the color {a} to the edge, which gives rise to a natural generalization of majestic colorings. For a positive integer k, let P([k]) be the power set of the set [k] = {1, 2, . . . , k} and let P∗([k]) = P([k]) − {∅} be the set of nonempty subsets of [k]. For each integer t with 1 ≤ t < k, let Pt([k]) be the set of t-element subsets of P([k]). For an edge coloring c : E(G) → Pt([k]) of a graph G, where adjacent edges may be colored the same, the vertex coloring c/ : V (G) → P∗([k]) is defined by c/(v) to be the union of the colors of edges incident with the vertex v in G. If c/ is a proper vertex coloring of G, then c is a majestic t-tone k-coloring of G. For a fixed positive integer t, the minimum positive integer k for which a graph G has a majestic t-tone k-coloring is the majestic t-tone index majt(G) of G. In particular, a majestic 1-tone k-coloring is a majestic k-coloring and the majestic 1-tone index of a graph G is the majestic index of G.

First, our emphasis is on the majestic 2-tone colorings in graphs. The values of maj2(G) are determined for several well-known classes of graphs G. We study the re- lationship between maj2(G ∨ K1) and maj2(G) where G ∨ K1 is the join of a graph G and K1. We then investigate the majestic t-tone indexes of connected graphs for t ≥ 2 in general with main emphasis on connected bipartite graphs. It is shown that if G is a connected bipartite graph of order at least 3, then majt(G) = t + 1 or majt(G) = t + 2 for each positive integer t. All trees, unicyclic bipartite graphs and connected bipartite graphs of cycle rank 2 having majestic t-tone index t + 1 have been characterized.

We also investigate the majestic t-tone indices of connected bipartite graphs having large cycles. In particular, we consider 2-connected bipartite graphs with large cycles. It is shown that (i) if G is a 2-connected bipartite graph of sufficiently large order n whose longest cycles have length £ where n − 5 ≤ £ ≤ n and t ≥ 2 is an integer, then majt(G) = t + 1 and (ii) there is a 2-connected bipartite graph F of sufficiently large order n whose longest cycles have length n − 6 and maj2(F ) = 4. Furthermore, it is shown for integers k, t ≥ 2 that there exists a k-connected bipartite graph G such that majt(G) = t + 2. Other results and open questions are also presented.

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Dissertation-Open Access

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