The Tur\'{a}n number of book graphs

Abstract

Given a graph $H$ and a positive integer $n,$ the Tur\'{a}n number of $H$ for the order $n,$ denoted ${\rm ex}(n,H),$ 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. Bollob\'{a}s and Erd\H{o}s initiated the research on the Tur\'{a}n number of book graphs in 1975. The two numbers ${\rm ex}(p+2,B_p)$ and ${\rm ex}(p+3,B_p)$ have been determined by Qiao and Zhan. In this paper we determine the numbers ${\rm ex}(p+4,B_p),$ ${\rm ex}(p+5,B_p)$ and ${\rm ex}(p+6,B_p),$ and characterize the corresponding extremal graphs for the numbers ${\rm ex}(n,B_p)$ with $n=p+2,\,p+3,\,p+4,\,p+5.$