Erdős, Gyárfás, and Pyber (1991) conjectured that every r-colored complete graph can be partitioned into at most r − 1 monochromatic components; this is a strengthening of a conjecture of Lovász (1975) and Ryser (1970) in which the components are only required to form a cover. An important partial result of Haxell and Kohayakawa (1995) shows that a partition into r monochromatic components is possible for sufficiently large r-colored complete graphs.We start by extending Haxell and Kohayakawa's result to graphs with large minimum degree, then we provide some partial analogs of their result for random graphs. In particular, we show that if p ≥ 27 log n n 1/3 , then a.a.s. in every 2coloring of G(n, p) there exists a partition into two monochromatic components, and for r ≥ 2 if p r log n n 1/r , then a.a.s. there exists an r-coloring of G(n, p) such that there does not exist a cover with a bounded number of components. Finally, we consider a random graph version of a classic result of Gyárfás (1977) about large monochromatic components in r-colored complete graphs. We show that if p = ω(1) n , then a.a.s. in every r-coloring of G(n, p) there exists a monochromatic component of order at least (1 − o(1)) n r−1 .

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Abstract. Hajnal and Szemerédi proved that every graph G with |G| = ks and δ(G) ≥ k(s − 1) contains k disjoint s-cliques; moreover this degree bound is optimal. We extend their theorem to directed graphs by showing that every directed graphOur result implies the Hajnal-Szemerédi Theorem, and its degree bound is optimal. We also make some conjectures regarding even more general results for multigraphs and partitioning into other tournaments. One of these conjectures is supported by an asymptotic result.

It is well known that in every r‐coloring of the edges of the complete bipartite graph K m , n there is a monochromatic connected component with at least false( m + n false) / r vertices. In this paper we study an extension of this problem by replacing complete bipartite graphs by bipartite graphs of large minimum degree. We conjecture that in every r‐coloring of the edges of an ( X , Y )‐bipartite graph with false| X false| = m , false| Y false| = n , δ ( X , Y ) > false( 1 − 1 / ( r + 1 ) false) n, and δ ( Y , X ) > false( 1 − 1 / false( r + 1 false) false) m, there exists a monochromatic component on at least false( m + n false) / r vertices (as in the complete bipartite graph). If true, the minimum degree condition is sharp (in that both inequalities cannot be made weak when m and n are divisible by r + 1). We prove the conjecture for r = 2 and we prove a weaker bound for all r ≥ 3. As a corollary, we obtain a result about the existence of monochromatic components with at least n/(r − 1) vertices in r‐colored graphs with large minimum degree.

Lehel conjectured that in every 2-coloring of the edges of K n , there is a vertex disjoint red and blue cycle which span V (K n ). Luczak, Rödl, and Szemerédi proved Lehel's conjecture for large n, Allen gave a different proof for large n, and finally Bessy and Thomassé gave a proof for all n.Balogh, Barát, Gerbner, Gyárfás, and Sárközy proposed a significant strengthening of Lehel's conjecture where K n is replaced by any graph G with δ(G) > 3n/4; if true, this minimum degree condition is essentially best possible. We prove that their conjecture holds when δ(G) > (3/4 + o(1))n. Our proof uses Szemerédi's regularity lemma along with the absorbing method of Rödl, Ruciński, and Szemerédi by first showing that the graph can be covered with monochromatic subgraphs having certain robust expansion properties. In 1973, Rosta [27] and independently, Faudree and Schelp [11] exactly determined the Ramsey number for all pairs of cycles, which gave an analog of Theorem 1.1 for cycles. Later, this was slightly refined by Faudree, Lesniak, and Schiermeyer [10] to give the following best possible result about long monochromatic cycles.