Monochromatic connected matchings in 2-edge-colored multipartite graphs

A matching $M$ in a graph $G$ is connected if all the edges of $M$ are in the same component of $G$. Following \L uczak,there have been many results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in $2$-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings.Comment: 29 pages, 2 figure

An improvement on Łuczak's connected matchings method

A connected matching in a graph G is a matching contained in a connected component of G. A well-known method due to Łuczak reduces problems about monochromatic paths and cycles in complete graphs to problems about monochromatic connected matchings in almost complete graphs. We show that these can be further reduced to problems about monochromatic connected matchings in complete graphs. We illustrate the potential of this new reduction by showing how it can be used to determine the 3-colour Ramsey number of long paths, using a simpler argument than the original one by Gyárfás, Ruszinkó, Sárközy, and Szemerédi (2007)

The bipartite Ramsey numbers BR(C8,C2n)BR(C_8, C_{2n})

For the given bipartite graphs $G_1,G_2,\ldots,G_t$, the multicolor bipartite Ramsey number $BR(G_1,G_2,\ldots,G_t)$ is the smallest positive integer $b$ such that any $t$-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$, colored with the $i$th color for some $1\leq i\leq t$. We compute the exact values of the bipartite Ramsey numbers $BR(C_8,C_{2n})$ for $n\geq2$

Multicolour bipartite Ramsey number of paths

The k-colour bipartite Ramsey number of a bipartite graph H is the least integer N for which every k-edge-coloured complete bipartite graph K-N,K-N contains a monochromatic copy of H. The study of bipartite Ramsey numbers was initiated over 40 years ago by Faudree and Schelp and, independently, by Gyarfas and Lehel, who determined the 2-colour bipartite Ramsey number of paths. Recently the 3-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gyarfas, Krueger, Ruszinko, and Sarkozy, in this paper we determine asymptotically the 4-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the k-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.ISSN:1097-1440ISSN:1077-892

Extremal problems on special graph colorings

In this thesis, we study several extremal problems on graph colorings. In particular, we study monochromatic connected matchings, paths, and cycles in 2-edge colored graphs, packing colorings of subcubic graphs, and directed intersection number of digraphs. In Chapter 2, we consider monochromatic structures in 2-edge colored graphs. A matching M in a graph G is connected if all the edges of M are in the same component of G. Following Łuczak, there are a number of results using the existence of large connected matchings in cluster graphs with respect to regular partitions of large graphs to show the existence of long paths and other structures in these graphs. We prove exact Ramsey-type bounds on the sizes of monochromatic connected matchings in 2-edge-colored multipartite graphs. In addition, we prove a stability theorem for such matchings, which is used to find necessary and sufficient conditions on the existence of monochromatic paths and cycles: for every fixed s and large n, we describe all values of n_1, ...,n_s such that for every 2-edge-coloring of the complete s-partite graph K_{n_1, ...,n_s} there exists a monochromatic (i) cycle C_{2n} with 2n vertices, (ii) cycle C_{at least 2n} with at least 2n vertices, (iii) path P_{2n} with 2n vertices, and (iv) path P_{2n+1} with 2n+1 vertices. Our results also imply for large n of the conjecture by Gyárfás, Ruszinkó, Sárkőzy and Szemerédi that for every 2-edge-coloring of the complete 3-partite graph K_{n,n,n} there is a monochromatic path P_{2n+1}. Moreover, we prove that for every sufficiently large n, if n = 3t+r where r in {0,1,2} and G is an n-vertex graph with minimum degree at least (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length {3, 4, 5, ..., 2t+r} of the same color, or there are cycles of every even length {4, 6, 8, ..., 2t+2} of the same color. This result is tight and implies the conjecture of Schelp that for every sufficiently large n, every (3n-1)-vertex graph G with minimum degree larger than 3|V(G)|/4, in each 2-edge-coloring of G there exists a monochromatic path P_{2n} with 2n vertices. It also implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for every positive integer n of the form n=3t+r where r in {0,1,2} and every n-vertex graph G with minimum degree at least 3n/4, in each 2-edge-coloring of G there exists a monochromatic cycle of length at least 2t+r. In Chapter 3, we consider a collection of special vertex colorings called packing colorings. For a sequence of non-decreasing positive integers S = (s_1, ..., s_k), a packing S-coloring is a partition of V(G) into sets V_1, ..., V_k such that for each integer i in {1, ..., k} the distance between any two distinct x,y in V_i is at least s_i+1. The smallest k such that G has a packing (1,2, ..., k)-coloring is called the packing chromatic number of G and is denoted by \chi_p(G). The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We show that for every fixed k and g at least 2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k, which answers the previous question in the negative. Moreover, we work towards the conjecture of Brešar, Klavžar, Rall and Wash that the packing chromatic number of 1-subdivision of subcubic graphs are bounded above by 5. In particular, we show that every subcubic graph is (1,1,2,2,3,3,k)-colorable for every integer k at least 4 via a coloring in which color k is used at most once, every 2-degenerate subcubic graph is (1,1,2,2,3,3)-colorable, and every subcubic graph with maximum average degree less than 30/11 is packing (1,1,2,2)-colorable. Furthermore, while proving the packing chromatic number of subcubic graphs is unbounded, we also consider improving upper bound on the independence ratio, alpha(G)/n, of cubic n-vertex graphs of large girth. We show that ``almost all" cubic labeled graphs of girth at least 16 have independence ratio at most 0.454. In Chapter 4, we introduce and study the directed intersection representation of digraphs. A directed intersection representation is an assignment of a color set to each vertex in a digraph such that two vertices form an edge if and only if their color sets share at least one color and the tail vertex has a strictly smaller color set than the head. The smallest possible size of the union of the color sets is defined to be the directed intersection number (DIN). We show that the directed intersection representation is well-defined for all directed acyclic graphs and the maximum DIN among all n vertex acyclic digraphs is at most 5n^2/8 + O(n) and at least 9n^2/16 + O(n)