The Turán number of book graphs

Abstract

Given a graph $H$ and a positive integer $n,$ the Turán number of $H$ for the order $n,$ denoted $\mathrm{ex}(n,H),$ is the maximum size of a simple graph of order $n$ not containing $H$ as a subgraph. The book with $p$ pages, denoted $B_p$, is the graph that consists of $p$ triangles sharing a common edge. Bollobás and Erdős initiated the research on the Turán number of book graphs in 1975. The two numbers $\mathrm{ex}(p+2,B_p)$ and $\mathrm{ex}(p+3,B_p)$ have been determined by Qiao and Zhan. In this paper we determine the numbers $\mathrm{ex}(p+4,B_p),$ $\mathrm{ex}(p+5,B_p)$ and $\mathrm{ex}(p+6,B_p),$ and characterize the corresponding extremal graphs for the numbers $\mathrm{ex}(n,B_p)$ with $n=p+2,p+3,p+4,p+5.$

16 pages, 7 figures