Extremal graphs for blow-ups of stars and paths

Abstract

The blow-up of a graph F is the graph obtained from F by replacing each edge in F by a clique of the same size where the new vertices of the cliques are all different. Given a forbidden graph H and a positive integer n, the extremal number, ex (n,H), is the maximum number of edges in a graph on n vertices that does not contain H as a subgraph. When forbidding multiple graphs, ex(n,H1,H2,…,Hk) denotes the maximum number of edges in an n-vertex graph with no subgraph Hi for 1≤i≤k. Erdös et al. and Chen et al. determined the extremal number of blow-ups of stars. Liu and Glebov determined the extremal numbers of the blow-ups of paths, cycles and a large class of trees respectively. In this paper we determine the extremal number and find the extremal graphs for both blow-up of stars and blow-up of paths, when n is sufficiently large.