For graphs F and H, we say F is Ramsey for H if every 2-coloring of the edges of F contains a monochromatic copy of H. The graph F is Ramsey H-minimal if there is no proper subgraph F ′ of F so that F ′ is Ramsey for H. Burr, Erdős, and Lovász defined s(H) to be the minimum degree of F over all Ramsey H-minimal graphs F . Define Ht,d to be a graph on t+ 1 vertices consisting of a complete graph...

An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs H and G , the anti-Ramsey number a r ( ) in maximum colors coloring so that there does not exist copy . Li et al. determined k -matchings complete bipartite graphs. Jin Zang showed uniqueness extremal coloring. In this paper, as generalization these results, we determine K n 1 … M -partite -un...

Unavoidable subgraphs have been widely studied in the context of Ramsey Theory. The research in this area focuses on highly structured graphs such as cliques, cycles, paths, stars, trees, and wheels. We propose to study maximum unavoidable subgraphs measuring the size in the number of edges. We computed maximum unavoidable subgraphs for graphs up to order nine via SAT solving and observed that ...

We show that limn→∞ r̂(F1,n, . . . , Fq,n, Fp+1, . . . , Fr)/n exists, where the bipartite graphs Fq+1, . . . , Fr do not depend on n while, for 1 ≤ i ≤ q, Fi,n is obtained from some bipartite graph Fi with parts V1 ∪ V2 = V (Fi) by duplicating each vertex v ∈ V2 (cv + o(1))n times for some real cv > 0. In fact, the limit is the minimum of a certain mixed integer program. Using the Farkas Lemma ...

The Regularity Lemma [16] is a powerful tool in Graph Theory and its applications. It basically says that every graph can be well approximated by the union of a constant number of random-looking bipartite graphs called regular pairs (see the definitions below). These bipartite graphs share many local properties with random bipartite graphs, e.g. most degrees are about the same, most pairs of ve...

For a graph H and an integer n, the Turán number ex(n,H) is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H . H is rdegenerate if every one of its subgraphs contains a vertex of degree at most r. We prove that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, ex(n,H) O(n2−1/r). This is tight for all values of...