Saturation numbers for Ramsey-minimal graphs

Given graphs $H_1, \dots, H_t$, a graph $G$ is $(H_1, \dots, H_t)$-Ramsey-minimal if every $t$-coloring of the edges of $G$ contains a monochromatic $H_i$ in color $i$ for some $i\in\{1, \dots, t\}$, but any proper subgraph of $G$ does not possess this property. We define $\mathcal{R}_{\min}(H_1, \dots, H_t)$ to be the family of $(H_1, \dots, H_t)$-Ramsey-minimal graphs. A graph $G$ is \dfn{$\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated} if no element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G$, but for any edge $e$ in $\overline{G}$, some element of $\mathcal{R}_{\min}(H_1, \dots, H_t)$ is a subgraph of $G + e$. We define $sat(n, \mathcal{R}_{\min}(H_1, \dots, H_t))$ to be the minimum number of edges over all $\mathcal{R}_{\min}(H_1, \dots, H_t)$-saturated graphs on $n$ vertices. In 1987, Hanson and Toft conjectured that $sat(n, \mathcal{R}_{\min}(K_{k_1}, \dots, K_{k_t}) )= (r - 2)(n - r + 2)+\binom{r - 2}{2}$ for $n \ge r$, where $r=r(K_{k_1}, \dots, K_{k_t})$ is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Toft's conjecture for sufficiently large $n$ was setteled in 2011, and is so far the only settled case. Motivated by Hanson and Toft's conjecture, we study the minimum number of edges over all $\mathcal{R}_{\min}(K_3, \mathcal{T}_k)$-saturated graphs on $n$ vertices, where $\mathcal{T}_k$ is the family of all trees on $k$ vertices. We show that for $n \ge 18$, $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_4)) =\lfloor {5n}/{2}\rfloor$. For $k \ge 5$ and $n \ge 2k + (\lceil k/2 \rceil +1) \lceil k/2 \rceil -2$, we obtain an asymptotic bound for $sat(n, \mathcal{R}_{\min}(K_3, \mathcal{T}_k))$.Comment: to appear in Discrete Mathematic

Coloring Graphs with Forbidden Minors

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. My research is motivated by the famous Hadwiger\u27s Conjecture from 1943 which states that every graph with no Kt-minor is (t − 1)-colorable. This conjecture has been proved true for t ≤ 6, but remains open for all t ≥ 7. For t = 7, it is not even yet known if a graph with no K7-minor is 7-colorable. We begin by showing that every graph with no Kt-minor is (2t − 6)- colorable for t = 7, 8, 9, in the process giving a shorter and computer-free proof of the known results for t = 7, 8. We also show that this result extends to larger values of t if Mader\u27s bound for the extremal function for Kt-minors is true. Additionally, we show that any graph with no K−8 - minor is 9-colorable, and any graph with no K=8-minor is 8-colorable. The Kempe-chain method developed for our proofs of the above results may be of independent interest. We also use Mader\u27s H-Wege theorem to establish some sufficient conditions for a graph to contain a K8-minor. Another motivation for my research is a well-known conjecture of Erdos and Lovasz from 1968, the Double-Critical Graph Conjecture. A connected graph G is double-critical if for all edges xy ∈ E(G), χ(G−x−y) = χ(G)−2. Erdos and Lovasz conjectured that the only double-critical t-chromatic graph is the complete graph Kt. This conjecture has been show to be true for t ≤ 5 and remains open for t ≥ 6. It has further been shown that any non-complete, double-critical, t-chromatic graph contains Kt as a minor for t ≤ 8. We give a shorter proof of this result for t = 7, a computer-free proof for t = 8, and extend the result to show that G contains a K9-minor for all t ≥ 9. Finally, we show that the Double-Critical Graph Conjecture is true for double-critical graphs with chromatic number t ≤ 8 if such graphs are claw-free

Two Ramsey-related Problems

Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. In this dissertation, we focus on studying the minimum number of edges of certain co-critical graphs. Given an integer r ≥ 1 and graphs G; H1; : : : ;Hr, we write → G (H1; : : : ;Hr) if every r-coloring of the edges of G contains a monochromatic copy of Hi in color i for some i ϵ {1; : : : ; r}. A non-complete graph G is (H1; : : : ;Hr)-co-critical if -/ \u3e (H1; : : : ;Hr), but G + uv → (H1; : : : ;Hr) for every pair of non-adjacent vertices u; v in G. Motivated in part by Hanson and Toft\u27s conjecture from 1987, we study the minimum number of edges over all (Kt; Tk)-co-critical graphs on n vertices, where Tk denotes the family of all trees on k vertices. We apply graph bootstrap percolation on a not necessarily Kt-saturated graph to prove that for all t ≥ 4 and k ≥ max{6, t}, there exists a constant c(t, k) such that, for all n ≥ (t - 1)(k - 1) + 1, if G is a (Kt; Tk)-co-critical graph on n vertices, then e(G) ≥ (4t-9/2 + 1/2 [K/2]) n - c _t, k). We then show that this is asymptotically best possible for all sufficiently large n when t ϵ {4, 5} and k ≥ 6. The method we developed may shed some light on solving Hanson and Toft\u27s conjecture, which is wide open. We also study Ramsey numbers of even cycles and paths under Gallai colorings, where a Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai k-coloring is a Gallai coloring that uses at most k colors. Given an integer k ≥ 1 and graphs H1, : : : ,Hk, the Gallai-Ramsey number GR(H1; : : : ;Hk) is the least integer n such that every Gallai k-coloring of the complete graph Kn contains a monochromatic copy of Hi in color i for some i ϵ {1; : : : ; k}. We completely determine the exact values of GR(H1; : : : ;Hk) for all k ≥ 2 when each Hi is a path or an even cycle on at most 13 vertices

Extremal problems in disjoint cycles and graph saturation

In this thesis, we tackle two main themes: sufficient conditions for the existence of particular subgraphs in a graph, and variations on graph saturation. Determining whether a graph contains a certain subgraph is a computationally difficult problem; as such, sufficient conditions for the existence of a given subgraph are prized. In Chapter 2, we offer a significant refinement of the Corradi-Hajnal Theorem, which gives sufficient conditions for the existence of a given number of disjoint cycles in a graph. Further, our refined theorem leads to an answer for a question posed by G. Dirac in 1963 regarding the existence of disjoint cycles in graphs with a certain connectivity. This answer comprises Chapter 3. In Chapter 4 we prove a result about equitable coloring: that is, a proper coloring whose color classes all have the same size. Our equitable-coloring result confirms a partial case of a generalized version of the much-studied Chen-Lih-Wu conjecture on equitable coloring. In addition, the equitable-coloring result is equivalent to a statement about the existence of disjoint cycles, contributing to our refinement of the Corradi-Hajnal Theorem. In Chapters 5 and 6, we move to the topic of graph saturation, which is related to the Turan problem. One imagines a set of n vertices, to which edges are added one-by-one so that a forbidden subgraph never appears. At some point, no more edges can be added. The Turan problem asks the maximum number of edges in such a graph; the saturation number, on the other hand, asks the minimum number of edges. Two variations of this parameter are studied. In Chapter 5, we study the saturation of Ramsey-minimal families. Ramsey theory deals with partitioning the edges of graphs so that each partition avoids the particular forbidden subgraph assigned to it. Our motivation for studying these families is that they provide a convincing edge-colored (Ramsey) version of graph saturation. We develop a method, called iterated recoloring, for using results from graph saturation to understand this Ramsey version of saturation. As a proof of concept, we use iterated recoloring to determine the saturation number of the Ramsey-minimal families of matchings and describe the assiociated extremal graphs. An induced version of graph saturation was suggested by Martin and Smith. In order to offer a parameter that is defined for all forbidden graphs, Martin and Smith consider generalized graphs, called trigraphs. Of particular interest is the case when the induced-saturated trigraphs in question are equivalent to graphs. In Chapter 6, we show that a surprisingly large number of families fall into this case. Further, we define and investigate another parameter that is a version of induced saturation that is closer in spirit to the original version of graph saturation, but that is not defined for all forbidden subgraphs