Sharp bounds for decomposing graphs into edges and triangles

Adam Blumenthal,Florian Pfender,Jan Volec,Bernard Lidický,Oleg Pikhurko,Yanitsa Pehova

doi:10.1017/s0963548320000358

Abstract

AbstractFor a real constant α, let $\pi _3^\alpha (G)$ 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 $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices.The extremal function $\pi _3^3(n)$ 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$\pi _3^3(n) \le (1/2 + o(1)){n^2}$. We extend their result by determining the exact value of $\pi _3^\alpha (n)$ 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 ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n.

Highlights

In recent progress on a problem of Gyori 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 K2 and K3 whose total number of vertices is at most (1/2 + o(1))n2, where Kr denotes the clique on272 A
The origins of this problem can be traced back to Erdos, 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
In this paper we show, by building upon the proof in [19], that for all large n it holds that π3(n) n2/2 + 1

Summary

Introduction

In recent progress on a problem of Gyori 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 K2 and K3 whose total number of vertices is at most (1/2 + o(1))n2, where Kr denotes the clique on. We will show that π3(G) π3(T2(n)) with equality if and only if G = T2(n) This claim can be directly derived from the result of Gyori [11, Theorem 1] that a graph with n vertices and t2(n) + k edges, where n → ∞ and k = o(n2), has at least k − O(k2/n2) edge-disjoint triangles. By Dirac’s theorem, this subgraph has a matching covering all but at most one vertex, that is, all edges between u and W except at most one are decomposed as triangles in Step 2. Note that the process does not reach i < n/2 for otherwise G has roughly at least (n/2) × (n/4) non-edges, which is a contradiction to G being δn2-close to Kn. Let Gs with |Gs| = s n/2 n0 be the graph for which the above process terminates. By comparing the costs of optimal decompositions, we conclude that G ∈ En

Extension to an arbitrary cost α