## Dissertations

12-2017

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

Dr. Ping Zhang

Dr. Gary Chartrand

Dr. Patrick Bennett

Dr. John Martino

#### Keywords

Graph theory, edge coloring, rainbow connection, information-transfer paths

#### Abstract

The Department of Homeland Security in the United States was created in 2003 in response to weaknesses discovered in the transfer of classied information after the September 11, 2001 terrorist attacks. While information related to national security needs to be protected, there must be procedures in place that permit access between appropriate parties. This two-fold issue can be addressed by assigning information-transfer paths between agencies which may have other agencies as intermediaries while requiring a large enough number of passwords and rewalls that is prohibitive to intruders, yet small enough to manage. Situations such as this can be represented by a graph whose vertices are the agencies and where two vertices are adjacent if there is direct access between them. Such graphs can then be studied by means of certain edge colorings of the graphs, where colors here refer to passwords. During the past decade, many research topics in graph theory have been introduced to deal with this type of problems. In particular, edge colorings of connected graphs have been introduced that deal with various ways every pair of vertices are connected by paths possessing some prescribed color condition.

Let G be an edge-colored connected graph, where adjacent edges may be colored the same. A path P is a rainbow path in an edge-colored graph G if no two edges of P are colored the same. An edge coloring c of a connected graph G is a rainbow coloring of G if every pair of distinct vertices of G are connected by a rainbow path in G. In this case, G is rainbow-connected. The minimum number of colors needed for a rainbow coloring of G is referred to as the rainbow connection number of G and is denoted by rc(G). A path P is a proper path in G if no two adjacent edges of P are colored the same. An edge coloring c of a connected graph G is a proper-path coloring of G if every pair of distinct vertices of G are connected by a proper path in G. If k colors are used, then c is referred to as a proper-path k-coloring. The minimum k for which G has a proper-path k-coloring is called the proper connection number pc(G) of G. In recent years, these two concepts have been studied extensively by many researchers. It has been observed that these two concepts model a communications network, where the goal is to transfer information in a secure manner between various law enforcement and intelligence agencies. Research on these two concepts has typically involved problems dealing with the minimum number of colors required for the graphical models of these communications networks to possess at least one desirable information-transfer path between each pair of agencies.

Looking at rainbow colorings and proper-path colorings in a different way brings up edge colorings that are intermediate to rainbow and proper-path colorings. Let G be a nontrivial connected edge-colored graph, where adjacent edges may be colored the same. A path P in G is a proper path if no two adjacent edges of P are colored the same and is a rainbow path if no two edges of P are colored the same. For an integer κ≥ 2, a path P in G is a κ-rainbow path if every subpath of P having length at most κ is a rainbow path. An edge coloring of G is a κ-rainbow coloring if every pair of distinct vertices of G are connected by a κ-rainbow path in G. The minimum number of colors for which G has a κ-rainbow coloring is called the κ-rainbow connection number of G. Thus, if G is a nontrivial connected graph whose longest paths have length ℓ, then

pc(G)=rc2(G)≥rc3(G)≥...≥rc ℓ(G)=rc(G).

We first investigate the 3-rainbow colorings in graphs and the relationships among the 3-rainbow connection numbers and the well-studied chromatic number, chromatic index, rainbow or proper connection numbers of graphs. Since every connected graph G contains a spanning tree T and rcκ (G) ≥ rcκ (T), the various connection numbers of trees play an important role in the study of general graphs. It is shown that.... (abstract continued in full dissertation)