Western Michigan University ScholarWorks at WMU Dissertations Graduate College 12-2017 Color-Connected Graphs and Information-Transfer Paths Stephen Devereaux Western Michigan University, [email protected] Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons Recommended Citation Devereaux, Stephen, "Color-Connected Graphs and Information-Transfer Paths" (2017). Dissertations. 3200. https://scholarworks.wmich.edu/dissertations/3200 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. Color-Connected Graphs and Information-Transfer Paths by Stephen Devereaux A dissertation submitted to the Graduate College in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mathematics Western Michigan University December 2017 Doctoral Committee: Ping Zhang, Ph.D., Chair Gary Chartrand, Ph.D. Patrick Bennett, Ph.D. John Martino, Ph.D. Stephen Hedetniemi, Ph.D. Color-Connected Graphs and Information-Transfer Paths Stephen Devereaux, Ph.D. Western Michigan University, 2017 The Department of Homeland Security in the United States was created in 2003 in response to weaknesses discovered in the transfer of classified 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 ap- propriate 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 firewalls 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 re- search 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 k ≥ 2, a path P in G is a k-rainbow path if every subpath of P having length at most k is a rainbow path. An edge coloring of G is a k-rainbow coloring if every pair of distinct vertices of G are connected by a k-rainbow path in G. The minimum number of colors for which G has a k-rainbow coloring is called the k-rainbow connection number of G. Thus, if G is a nontrivial connected graph whose longest paths have length `, then pc(G) = rc2(G) ≤ rc3(G) ≤ · · · ≤ rc`(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 rck(G) ≤ rck(T ), the various connection numbers of trees play an important role in the study of general graphs. It is shown that ? for a triple (a; b; c) of integers with 2 ≤ a ≤ b ≤ c, there exists a tree T with pc(T ) = a, rc3(T ) = b and rc(T ) = c if and only if a = b = c or 2 ≤ a < b ≤ minf2a − 1; cg and ? for a triple (a; b; m)of positive integers, there exists a tree T of size m with χ0(G) = a and rc3(T ) = b if and only if a ≤ b ≤ m such that a = b = m or 2 ≤ a < b ≤ minf2a − 1; mg. For each integer k ≥ 3, the values of rck(G) are determined for several well-known classes of graphs G. For example, if Ks;t is the complete bipartite graph 2 ≤ s ≤ t, ` is the length of a longest path in Ks;t and k is an integer with 2 ≤ k ≤ `, then 8 2 if k = 2 > p < s rck(Ks;t) = min t ; 3 if k = 3 > p : min s t ; 4 if 4 ≤ k ≤ `. If T is a tree of diameter at least k for some integer k ≥ 2, then 0 0 0 rck(T ) = maxfm(T ): T ⊆ T and diam(T ) = kg: With the aid of results on trees, upper bounds for rck(G) are established in terms of the maximum degree of G. For example, if G is a connected graph of order at least k +1 ≥ 4 and maximum degree ∆ ≥ 3, then 8 ∆[(∆−1)t−1] > if k = 2t ≥ 2 is even <> ∆−2 rck(G) ≤ > [(∆−1)t−1−1] : 1 + 2(∆ − 1) ∆−2 if k = 2t − 1 ≥ 3 is odd. We also establish sharp upper bounds for a connected graph in terms of its order. ? If G is a nontrivial connected graph of order n ≥ 3 that is not a tree such that the length of a longest path in G is `, then rck(G) ≤ n − 2 for all integers k with 2 ≤ k ≤ `. Furthermore, the upper bound n − 2 is best possible. ? Let G be a connected graph of order n ≥ 4 and size m ≥ n + 1 and let ` be the length of a longest path in G. If G does not contain K4 − e as a subgraph, then rck(G) ≤ n − 3 for all integers k with 2 ≤ k ≤ `. Furthermore, the upper bound n − 3 is best possible. If G is a nontrivial connected graph of diameter d, then rcd(G) ≥ d. Furthermore, if G is a nontrivial tree of order n, then rcd(G) = n − 1 and so rcd(G) − diam(G) can be arbitrarily large. On the other hand, if G is a connected graph of order n ≥ 3 and diameter d ≥ 2 that is not tree, then d ≤ rcd(G) ≤ n − 2. It is shown that for each triple (d; k; n) of integers with 2 ≤ d ≤ k ≤ n − 1, there exists a connected graph G of order n that is not a tree such that diam(G) = d and rcd(G) = k if and only if k 6= n − 1. One of well-known areas of research in graph theory involves the Hamiltonian prop- erties of graphs. A Hamiltonian cycle in a graph G is a cycle containing every vertex of G and a graph having a Hamiltonian cycle is a Hamiltonian graph. A Hamiltonian path in a graph G is a path containing every vertex of G. A graph G is Hamiltonian-connected if G contains a Hamiltonian u − v path for every pair u; v of distinct vertices of G. In a rainbow coloring or a proper-path coloring of a connected graph G, every two vertices u and v of G are connected by a rainbow u − v path or a proper u − v path and there is no condition on what the length of such a path must be. For certain graphs G, however, it is natural to ask whether there may exist an edge coloring of G using a certain number of colors such that every two vertices of G are connected by a rainbow path or proper path of a prescribed length. For a Hamiltonian-connected graph G, an edge coloring c is called a Hamiltonian- connected rainbow coloring if every two vertices of G are connected by a rainbow Hamil- tonian path in G.