Kyle Kolasinski

On Monochromatic Subgraphs of Edge-Colored Complete Graphs

Abstract In a red-blue coloring of a nonempty graph, every edge is colored red or blue. If the resulting edge-colored graph contains a nonempty subgraph G without isolated vertices every edge of which is colored the same, then G is said to be monochromatic. For two nonempty graphs G and H without isolated vertices, the mono- chromatic Ramsey number mr(G,H) of G and H is the minimum integer n such that every red-blue coloring of Kn results in a monochromatic G or a monochromatic H. Thus, the standard Ramsey number of G and H is bounded below by mr(G,H). The monochromatic Ramsey numbers of graphs belonging to some common classes of graphs are studied. We also investigate another concept closely related to the standard Ram- sey numbers and monochromatic Ramsey numbers of graphs. For a fixed integer n ≥ 3, consider a nonempty subgraph G of order at most n con- taining no isolated vertices. Then G is a common monochromatic subgraph of Kn if every red-blue coloring of Kn results in a monochromatic copy of G. Furthermore, G is a maximal common monochromatic subgraph of Kn if G is a common monochromatic subgraph of Kn that is not a proper sub- graph of any common monochromatic subgraph of Kn. Let S(n) and S*(n) be the sets of common monochromatic subgraphs and maximal common monochromatic subgraphs of Kn, respectively. Thus, G ∈ S(n) if and only if R(G,G) = mr(G,G) ≤ n. We determine the sets S(n) and S*(n) for 3 ≤ n ≤ 8.

VERTEX RAINBOW COLORINGS OF GRAPHS

In a properly vertex-colored graph G, a path P is a rainbow path if no two vertices of P have the same color, except possibly the two end-vertices of P. If every two vertices of G are connected by a rainbow path, then G is vertex rainbow-connected. A proper vertex coloring of a connected graph G that results in a vertex rainbow-connected graph is a vertex rainbow coloring of G. The minimum number of colors needed in a vertex rainbow coloring of G is the vertex rainbow connection number vrc(G) of G. Thus if G is a connected graph of order n ≥ 2, then 2 ≤ vrc(G) ≤ n. We present characterizations of all connected graphs G of order n for which vrc(G) ∈ {2,n−1,n} and study the relationship between vrc(G) and the chromatic number χ(G) of G. For a connected graph G of order n and size m, the number m − n + 1 is the cycle rank of G. Vertex rainbow connection numbers are determined for all connected graphs of cycle rank 0 or 1 and these numbers are investigated for connected graphs of cycle rank 2.

HAMILTONIAN-COLORED POWERS OF STRONG DIGRAPHS

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power D k of D is that digraph having vertex set V (D) with the property that (u,v) is an arc of D k if the directed distance ~ dD(u,v) from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph D k is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph D k is distance-colored if each arc (u,v) of D k is assigned the color i where i = ~ dD(u,v). The digraph D k is Hamiltonian-colored if D k contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which D k is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle ~ Cn of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph Dk such that hce(Dk) = k with the added property that every properly colored Hamiltonian cycle in the kth power of Dk must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D