R(W5 , K5) = 27

The two-color Ramsey number R(G , H) is defined to be the smallest integer n such that any graph F on n vertices contains either a subgraph isomorphic to G or the complement of F contains a subgraph isomorphic to H. Ramsey numbers serve to quantify many of the existing theorems of Ramsey theory, which looks at large combinatorial objects for certain given smaller combinatorial objects that must be present. In 1989 George R. T. Hendry presented a table of two-color Ramsey numbers R(G , H) for all pairs of graphs G and H having at most five vertices. This table left seven unsolved cases, of which three have since been solved. This thesis eliminates one of the remaining four cases, R(W5 , K5), where a K5 is the complete graph on five vertices and a W5 is a wheel of order 5, which can be pictured as a wheel having four spokes or as a cycle of length 4 having all four vertices adjacent to a central vertex. In this thesis we show R(W5, K5) to be equal to 27, utilizing a combinatorial approach with significant computations. Specifically we use a technique developed by McKay and Radziszowski to effectively glue together smaller graphs in an effort to prove exhaustively that no graph having 27 vertices exists that does not contain an independent set on five vertices or a subgraph isomorphic to W5. The previous best bounds for this case were 27 \u3c= R( W_5 , K_5 ) \u3c= 29

Hadwiger Numbers and Gallai-Ramsey Numbers of Special Graphs

This dissertation explores two separate topics on graphs. We first study a far-reaching generalization of the Four Color Theorem. Given a graph G, we use chi(G) to denote the chromatic number; alpha(G) the independence number; and h(G) the Hadwiger number, which is the largest integer t such that the complete graph K_t can be obtained from a subgraph of G by contracting edges. Hadwiger\u27s conjecture from 1943 states that for every graph G, h(G) is greater than or equal to chi(G). This is perhaps the most famous conjecture in Graph Theory and remains open even for graphs G with alpha(G) less than or equal to 2. Let W_5 denote the wheel on six vertices. We establish more evidence for Hadwiger\u27s conjecture by proving that h(G) is greater than or equal to chi(G) for all graphs G such that alpha(G) is less than or equal to 2 and G does not contain W_5 as an induced subgraph. Our second topic is related to Ramsey theory, a field that has intrigued those who study combinatorics for many decades. Computing the classical Ramsey numbers is a notoriously difficult problem, leaving many basic questions unanswered even after more than 80 years. We study Ramsey numbers under Gallai-colorings. A Gallai-coloring of a complete graph is an edge-coloring such that no triangle is colored with three distinct colors. Given a graph H and an integer k at least 1, the Gallai-Ramsey number, denoted GR_k(H), is the least positive integer n such that every Gallai-coloring of K_n with at most k colors contains a monochromatic copy of H. It turns out that GR_k(H) is more well-behaved than the classical Ramsey number R_k(H), though finding exact values of GR_k(H) is far from trivial. We show that for all k at least 3, GR_k(C_{2n+1}) = n2^k+1 where n is 4, 5, 6 or 7, and GR_k(C_{2n+1}) is at most (n ln n)2^k-(k+1)n+1 for all n at least 8, where C_{2n+1} denotes a cycle on 2n+1 vertices

Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

On small Mixed Pattern Ramsey numbers

We call the minimum order of any complete graph so that for any coloring of the edges by $k$ colors it is impossible to avoid a monochromatic or rainbow triangle, a Mixed Ramsey number. For any graph $H$ with edges colored from the above set of $k$ colors, if we consider the condition of excluding $H$ in the above definition, we produce a \emph{Mixed Pattern Ramsey number}, denoted $M_k(H)$. We determine this function in terms of $k$ for all colored $4$-cycles and all colored $4$-cliques. We also find bounds for $M_k(H)$ when $H$ is a monochromatic odd cycles, or a star for sufficiently large $k$. We state several open questions.Comment: 16 page

Improved bounds on the multicolor Ramsey numbers of paths and even cycles

We study the multicolor Ramsey numbers for paths and even cycles, $R_k(P_n)$ and $R_k(C_n)$, which are the smallest integers $N$ such that every coloring of the complete graph $K_N$ has a monochromatic copy of $P_n$ or $C_n$ respectively. For a long time, $R_k(P_n)$ has only been known to lie between $(k-1+o(1))n$ and $(k + o(1))n$. A recent breakthrough by S\'ark\"ozy and later improvement by Davies, Jenssen and Roberts give an upper bound of $(k - \frac{1}{4} + o(1))n$. We improve the upper bound to $(k - \frac{1}{2}+ o(1))n$. Our approach uses structural insights in connected graphs without a large matching. These insights may be of independent interest