## Date of Award

6-2012

## Degree Name

Doctor of Philosophy

## Department

Mathematics

## First Advisor

Dr. Ping Zhang

## Second Advisor

Dr. Gary Chartrand

## Third Advisor

Dr. Garry Johns

## Fourth Advisor

Dr. Allen Schwenk

## Abstract

Historically, the subject of graph colorings has been the most popular research area in graph theory. There are many problems in mathematics and in real life that can be represented by a graph and whose solution involves finding a specific coloring of this graph. Our research consists of two parts: (1) combinatorial problems and vertex colorings and (2) distance-defined colorings. In this research, we show that certain combinatorial puzzles and problems can be placed in a graph coloring setting and graph colorings can be defined in terms of distance in graphs that are useful in applications.

Two vertices u and v in a nontrivial connected graph *G *are twins if u and v have the same neighbors in *V *( G) - { u, v}. If u and v are adjacent, they are referred to as true twins; while if u and v are nonadjacent, they are false twins. For a positive integer *k, *let c: *V(G) *→ Zk be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c' : *V(G) *-+ Zk defined by c'(v) = ∑_{uϵN[v]} c(u) for each v ϵ *V(G), *where N[v] is the closed neighborhood of v. Then c is called a closed modular k-coloring if c' (*u) ≠ * c' (*v) *in Zk for all pairs *u, v *of adjacent vertices that are not true twins. The minimum positive integer *k *for which *G *has a closed modular k-coloring is the closed modular chromatic number mc¯(G) of G. These concepts were inspired by two combinatorial problems.

For an ordered set *W *= { w_{1}, w_{2}, ... , w_{k}} of *k *distinct vertices in a nontrivial connected graph *G , *the metric code of a vertex v of

*G*with respect to

*W*is the k-vector

code(v) = (d(v,w_{1}),d(v,w_{2}), .. · ,d(v,w_{k}))

where *d ( *v, w

_{1}) is the distance between v and Wi for 1 ≤ i ≤ k

**The set**

*.**W*is a local metric set of

*G*if code(

*u) ≠*code(

*v)*for every pair

*u, v*of adjacent vertices of

*G.*The minimum positive integer

*k*for which

*G*has a local metric k-set is the local metric dimension lmd(G) of

*G.*A local metric set of

*G*of cardinality lmd(G) is a local metric basis of

*G.*These concepts were inspired by the well-studied concepts of metric sets and metric dimension in graphs and their applications.

## Access Setting

Dissertation-Open Access

## Recommended Citation

Phinezy, Bryan, "Variations on a Graph Coloring Theme" (2012). *Dissertations*. 3400.

https://scholarworks.wmich.edu/dissertations/3400