Extremal problems on cycle structure and colorings of graphs

In this Thesis, we consider two main themes: conditions that guarantee diverse cycle structure within a graph, and the existence of strong edge-colorings for a specific family of graphs. In Chapter 2 we consider a question closely related to the Matthews-Sumner conjecture, which states that every 4-connected claw-free graph is Hamiltonian. Since there exists an infinite family of 4-connected claw-free graphs that are not pancyclic, Gould posed the problem of characterizing the pairs of graphs, {X,Y}, such that every 4-connected {X,Y}-free graph is pancyclic. In this chapter we describe a family of pairs of graphs such that if every 4-connected {X,Y}-free graph is pancyclic, then {X,Y} is in this family. Furthermore, we show that every 4-connected {K_(1,3),N(4,1,1)}-free graph is pancyclic. This result, together with several others, completes a characterization of the family of subgraphs, F such that for all H in ∈, every 4-connected {K_(1,3), H}-free graph is pancyclic. In Chapters and 4 we consider refinements of results on cycles and chorded cycles. In 1963, Corrádi and Hajnal proved a conjecture of Erdös, showing that every graph G on at least 3k vertices with minimum degree at least 2k contains k disjoint cycles. This result was extended by Enomoto and Wang, who independently proved that graphs on at least 3kvertices with minimum degree-sum at least 4k - 1 also contain k disjoint cycles. Both results are best possible, and recently, Kierstead, Kostochka, Molla, and Yeager characterized their sharpness examples. A chorded cycle analogue to the result of Corrádi and Hajnal was proved by Finkel, and a similar analogue to the result of Enomoto and Wang was proved by Chiba, Fujita, Gao, and Li. In Chapter 3 we characterize the sharpness examples to these statements, which provides a chorded cycle analogue to the characterization of Kierstead et al. In Chapter 4 we consider another result of Chiba et al., which states that for all integers r and s with r + s ≥ 1, every graph G on at least 3r + 4s vertices with ẟ(G) ≥ 2r+3s contains r disjoint cycles and s disjoint chorded cycles. We provide a characterization of the sharpness examples to this result, which yields a transition between the characterization of Kierstead et al. and the main result of Chapter 3. In Chapter 5 we move to the topic of edge-colorings, considering a variation known as strong edge-coloring. In 1990, Faudree, Gyárfás, Schelp, and Tuza posed several conjectures regarding strong edge-colorings of subcubic graphs. In particular, they conjectured that every subcubic planar graph has a strong edge-coloring using at most nine colors. We prove a slightly stronger form of this conjecture, showing that it holds for all subcubic planar loopless multigraphs