On the size of $(K_t, K_{1,k})$-co-critical graphs

Abstract

Given graphs G, H 1 , H 2 , we write G → (H 1 , H 2 ) if every {red, blue}-coloring of the edges of G contains a red copy of H 1 or a blue copy ofMotivated by a conjecture of Hanson and Toft from 1987, we study the minimum number of edges over all (K t , K 1,k )-cocritical graphs on n vertices. We prove that for all t ≥ 3 and k ≥ 3, there exists a constantFurthermore, this linear bound is asymptotically best possible when t ∈ {3, 4, 5} and all k ≥ 3 and n ≥ (2t − 2)k + 1. It seems non-trivial to construct extremal (K t , K 1,k )-co-critical graphs for t ≥ 6. We also obtain the sharp bound for the size of (K 3 , K 1,3 )-co-critical graphs on n ≥ 13 vertices by showing that all such graphs have at least 3n − 4 edges.