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

TL;DR

This paper determines the Turán numbers for book graphs with specific parameters and characterizes the extremal graphs for these cases, advancing the understanding of graph structures avoiding certain subgraphs.

Contribution

It extends previous work by explicitly calculating Turán numbers for larger graph orders and characterizing extremal graphs for these cases.

Findings

Determined ex(p+4, B_p), ex(p+5, B_p), ex(p+6, B_p).

Characterized extremal graphs for ex(n, B_p) with n=p+2 to p+5.

Builds on classical results by Bollobás and Erdős, and recent work by Qiao and Zhan.

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 $n=p+2,\,p+3,\,p+4,\,p+5.$