Date of Award

8-2026

Degree Name

Doctor of Philosophy

Department

Mathematics

First Advisor

Ping Zhang, Ph.D.

Second Advisor

Gary Chartrand, Ph.D.

Third Advisor

Jim Zhu, Ph.D.

Fourth Advisor

Dinesh Sarvate, Ph.D.

Keywords

Graph decomposition, Ramsey chain, Ramsey index, Ramsey theory, red-blue edge coloring

Abstract

The research in this work deals v-rith decomposition of graphs and Ramsey theory, two popular areas of research in graph theory due to numerous fascinating problems and the sheer mathema ical beauty of the subjects. A decomposition {G1 , G2 , ... , Gk} of a graph G into subgraphs of G is ascending if the subgraph Ci is isomorphic to a proper subgraph of G;+1 for i = 1, 2, ... , k - l. The well-known and long-standing of Ascending Subgraph Decomposition Conjecture states that every graph has an ascending subgraph decomposition, that is, the subgraphs (links) in the decomposition sequence are pairwise edge-disjoint in the given graph, monochromatic, and every link is a subgraph of all succeeding links in the sequence. One of the major topics in graph theory involving edge colorings takes place in Ramsey theory where typically for each red-blue edge coloring of a given graph, one of two prescribed monochromat ic subgraph occurs. Ramsey theory is a combinatorial concept that deals with the struct ural analysis of a complex graphical system to assert the existence of prescribed unavoidable patterns or substructures v-rithin complete randomness. In graph theory, this translates to the k-edge coloring (k E N) of a complete graph of sufficiently large order such that any such coloring preserves the existence of a monochromatic copy of some graph of smaller order. The Ramsey number of a sequence of graphs (G1, G2, .. . , Gk) , denoted by R(G1 , G2, ... , Gk ), is the minimum positive integer n such that any k - edge coloring of the complete graph K n of order n produces a monochromatic copy of Ci for some i E {l , 2, ... , k }.

In this research study, we investigate the concept of a Ramsey chain which involves the existence of pairv-rise edge-disjoint monochromatic subgraphs in a red-blue edge coloring of a given graph that satisfies the Ascending Subgraph Decomposition (ASD) condit ions. The color-specific Ramsey index of a graph G, denoted by ARc(G), is t he ma.ximum length of a Ramsey chain for a given red-blue edge coloring c of G . The Ramsey index of a graph, denoted by AR(G), is t he minimum taken over all ARc(G), for every red-blue edge coloring c of G ( that is, AR( G) = min { ARc ( G) : for all red-blue edge coloring c of G}) . The concepts of Ramsey chain and Ramsey index provide a different point of view on the Ascending Subgraph Decomposition Conjecture by means of edge-colorings of graphs. The primary goals of this work is to study the structures of Ramsey chains in several well-known classes of graphs, determine the Ramsey indices of these graphs, and investigate Ramsey chains in graphs with prescribed restrictions. Results and open questions are presented in this area of research.

Access Setting

Dissertation-Open Access

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