Triangle factors of graphs without large independent sets and of weighted graphs

Abstract

ABSTRACTThe classical Corrádi‐Hajnal theorem claims that every n‐vertex graph G with contains a triangle factor, when . In this paper we present two related results that both use the absorbing technique of Rödl, Ruciński and Szemerédi. Our main result determines the minimum degree condition necessary to guarantee a triangle factor in graphs with sublinear independence number. In particular, we show that if G is an n‐vertex graph with and , then G has a triangle factor and this is asymptotically best possible. Furthermore, it is shown for every r that if every linear size vertex set of a graph G spans quadratically many edges, and , then G has a Kr‐factor for n sufficiently large. We also propose many related open problems whose solutions could show a relationship with Ramsey‐Turán theory.Additionally, we also consider a fractional variant of the Corrádi‐Hajnal Theorem, settling a conjecture of Balogh‐Kemkes‐Lee‐Young. Let and . We call a triangle t‐heavy if the sum of the weights on its edges is more than 3t. We prove that if and w is such that for every vertex v the sum of w(e) over edges e incident to v is at least , then there are vertex disjoint heavy triangles in G. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 669–693, 2016