The degree Ramsey number of a graph G, denoted R∆(G; s), is min{∆(H) : H s → G}, where H s → G means that every s-edge-coloring of H contains a monochromatic copy of G. The closed k-blowup of a graph is obtained by replacing every vertex with a clique of size k and every edge with a complete bipartite graph where both partite sets have size k. We prove that there is a function f such that R∆(G;...

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...

For a given r-uniform hypergraph F we study the largest blow-up of F which can be guaranteed in every large r-uniform hypergraph with many copies of F . For graphs this problem was addressed by Nikiforov, who proved that every n-vertex graph that contains Ω(n`) copies of the complete graph K` must contain a complete `-partite graph with Ω(logn) vertices in each class. We give another proof of N...

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let Tn,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that Tn,r is the only extremal graph for complete graph Kr+1. Erdős et al. (1995) determined the ...

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...

A well-known special case of a conjecture attributed to Ryser (actually appeared in the thesis of Henderson [7]) states that k-partite intersecting hypergraphs have transversals of at most k−1 vertices. An equivalent form of the conjecture in terms of coloring of complete graphs is formulated in [1]: if the edges of a complete graph K are colored with k colors then the vertex set of K can be co...

In this paper, it is shown that the graph obtained by overlapping the cycle and the complete tripartite graph at an edge is uniquely determined by its chromatic polynomial. ) 3 ( ≥ m Cm 2 , 2 , 2 K Let G be a finite graph with neither loops nor multiple edges and let ) ; ( λ G P denote its chromatic polynomial. Then G is said to be chromatically unique if ) ; ( ) ; ( λ λ G Y P = implies that Y ...

A graph G is r-equitably k-colorable if its vertex set can be partitioned into k independent sets, any two of which differ in size by at most r. The r-equitable chromatic threshold of a graph G, denoted by χr=(G), is the minimum k such that G is r-equitably k -colorable for all k ≥ k. Let G × H denote the Kronecker product of graphs G and H. In this paper, we completely determine the exact valu...

By using the Szemerédi Regularity Lemma [10], Alon and Sudakov [1] recently extended the classical Andrásfai-Erdős-Sós theorem [2] to cover general graphs. We prove, without using the Regularity Lemma, that the following stronger statement is true. Given any (r+1)-partite graph H whose smallest part has t vertices, there exists a constant C such that for any given ε > 0 and sufficiently large n...