Extremal graphs for edge blow-up of lollipops

Abstract

Given a graph H and an integer p (p ≥ 2), the edge blow-up H p+1 of H is the graph obtained from replacing each edge in H by a clique of order (p + 1), where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. Very recently some substantial progress of the extremal graphs for H p+1 of larger p has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when p ≥ 3 and the exact Turán numbers for edge blow-up of all non-bipartite graphs when p ≥ χ(H) + 1 has been determined by Yuan (2022), where χ(H) is the chromatic number of H. A lollipop C k, ℓ is the graph obtained from a cycle C k by appending a path P ℓ+1 to one of its vertices. In this paper, we consider the extremal graphs for C p+1 k, ℓ of the rest cases p = 2 and p = 3.