Sharp bounds for decomposing graphs into edges and triangles

Abstract

For a real constant α, let πα(G) be the minimum of twice the number of K₂’s plus α times the number of K₃’s over all edge decompositions of G into copies of K₂ and K₃, where Kr denotes the complete graph on r vertices. Let πα(n) be the maximum of πα(G) over all graphs G with n vertices. <br/>

The extremal function π ³(n) was first studied by Győri and Tuza (Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova (Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that π ³(n) (1/2+o(1))n2 . We extend their result by determining the exact value of πα(n) and the set of extremal graphs for all α and sufficiently large n. In particular, we show for α 3 that Kn and the complete bipartite graph Kln/₂J,₁n/₂l are the only possible extremal examples for large n.