Committee:
Dr. Ping Zhang, Chair
Dr. Gary Chartrand
Dr. Garry Johns
Dr. Allen Schwenk
Dr. Arthur White

Date: Friday, May 18, 2012 2:00 p.m. to 4:00 p.m.
6625 Everett Tower

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 will 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.

For a positive integer k, let c : V (G) → ?k be a vertex coloring where adjacent vertices may be assigned the same color. The coloring c induces another vertex coloring c': V (G) → ?k defined by

c'(v) = ∑ c(u)
u∈N [v]

For each v ∈ 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 ?k for all pairs u, v of adjacent vertices that are not true twins. (over)
For an ordered set W = {w1 , w2 , … , wk } of k distinct vertices in a nontrivial con-
nected graph G, the metric code of a vertex v of G with respect to W is the k-vector

code(v) = (d(v, w1), d(v, w2), … , d(v, wk ))

where d(v, wi ) 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 motivation of these colorings is discussed. Results, applications, and open questions for these colorings is presented.

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