Extremal Graphs for Blow-Ups of Keyrings

Abstract

The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Given a 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. A keyring $$C_s(k)$$ is a $$(k+s)$$ -edge graph obtained from a cycle of length k by appending s leaves to one of its vertices. This paper determines the extremal number and finds the extremal graphs for the blow-ups of keyrings $$C_s(k)$$ ( $$k\ge 3$$ , $$s\ge 1$$ ) when n is sufficiently large. For special cases when $$k=0$$ or $$s=0$$ , the extremal number of the blow-ups of the graph $$C_s(0)$$ (a star) has been determined by Erdos et al. (J Comb Theory Ser B 64:89–100, 1995) and Chen et al. (J Comb Theory Ser B 89: 159–171, 2003), while the extremal number and extremal graphs for the blow-ups of the graph $$C_0(k)$$ (a cycle) when n is sufficiently large has been determined by Liu (Electron J Combin 20: P65, 2013).