Extremal graphs and multigraphs with two weighted colours

Abstract

We study the extremal properties of coloured multigraphs H, whose edge set is the union of two simple graphs Hr and Hb (thought of as red and blue edges) on the same vertex set. Let 0 ≤ p ≤ 1 and let q = 1 - p. The extremal problem considered here, for a given fixed H, is to find the maximum weight p|E(Gr)| + q|E(Gb)| of large coloured multigraphs G that do not contain H as a subgraph. In fact, motivated by applications (typically to the study of hereditary properties by means of Szemerédi's Lemma), we consider the maximum restricted to those G whose underlying graph is complete - that is, every pair of vertices is joined by at least one edge. We describe some basic features of the extremal function in general; in particular it is shown that, for any class of forbidden graphs, the extremal function always has a finite description. We then look at some examples, including a detailed study of the case Hr = K 3, 3 and Hb = Hr - 2K3. Our approach is that of classical extremal graph theory (and, as far as general properties are concerned, mirrors the work of Brown, Erdös and Simonovits on directed graphs). The recent work of Lovász and his co-authors on graph sequences might offer an alternative approach.