Decomposing Graphs into Edges and Triangles

Abstract

We prove the following 30-year old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C 1 , . . . , C ℓ of orders two and three such that |C 1 | + · · · + |C ℓ | ≤ (1/2 + o(1))n 2 . This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n 2 /4.

Sharp bounds for decomposing graphs into edges and triangles

“…In this paper we show, by building upon the proof in [19], that for all large n it holds in fact that π 3 (n) n 2 /2 + 1. Moreover, if a graph G of order n attains π 3 (n), then G is the complete graph K n or the Turán graph T 2 (n).…”

“…Győri and Tuza [16] showed that π 3 (n) 9n 2 /16. This was the best known bound until recently, when using the celebrated flag algebra method of Razborov [24], Král' , Lidický, Martins and Pehova [19] proved the asymptotic version of Tuza's conjecture. Theorem 1.3 (Král' et al [19]).…”

“…For this purpose, we start by showing that all almost π 3 -extremal graphs contain almost no copies of the three graphs in Figure 1 (which are obtained by taking the unlabelled versions of the corresponding graphs in − → F ). This is achieved by the following lemma, which builds on the results from [19]. Proof.…”

“…In recent progress on a problem of Győri and Tuza [27], Král' , Lidický, Martins and Pehova [19] proved via flag algebras that the edges of any n-vertex graph can be decomposed into copies of K 2 and K 3 whose total number of vertices is at most (1/2 + o(1))n 2 , where K r denotes the clique on r vertices. The origins of this problem can be traced back to Erdős, Goodman and Pósa [10], who considered the problem of minimizing the total number of cliques in an edge decomposition of an arbitrary n-vertex graph.…”

“…This paper is organized as follows. In Section 2 we give an outline of the proof of Theorem 1.3 from [19] that we build on. Theorem 1.5 is proved in Section 3.…”