#### Title

#### 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 *□ *K*_{2} 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 P* _{t}*([

*k*]) be the set of

*t*-element subsets of P([

*k*]). For an edge coloring

*c*

*:*

*E*(

*G*) → P

*([*

_{t}*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 maj

*t*(

*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** *∨ *K*_{1}) and maj2(*G*) where *G *∨ *K*_{1} is the join of a graph *G *and *K*_{1}. 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 maj*t*(*G*) = *t *+ 1 or maj*t*(*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 maj*t*(*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 maj*t*(*G*) = *t *+ 2. Other results and open questions are also presented.

#### Access Setting

Dissertation-Open Access

#### Recommended Citation

Hart, Ian, "Induced Graph Colorings" (2018). *Dissertations*. 3309.

https://scholarworks.wmich.edu/dissertations/3309