Tur\'{a}n number of books in non-bipartite graphs

TL;DR

This paper determines the exact Turán numbers and characterizes extremal non-bipartite graphs avoiding book graphs $B_{r+1}$ for large $n$, revealing a stark difference from bipartite cases.

Contribution

It provides the exact Turán numbers and characterizations of extremal graphs for non-bipartite $B_{r+1}$-free graphs, extending classical results to a new non-bipartite setting.

Findings

Exact Turán numbers for non-bipartite $B_{r+1}$-free graphs.

Complete characterization of extremal graphs for large $n$.

Different behaviors of Turán numbers for $r=0$ and $r eq0$.

Abstract

Let $\mathrm{ex}(n, H)$ be the Tur\'{a}n number of $H$ for a given graph $H$. A graph is color-critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if $H$ is color-critical with $\chi(H)=k+1$, then there exists an $n_0(H)$ such that ex$(n, H)=e(T_{n,k})$ and the Tur\'{a}n graph $T_{n,k}$ is the only extremal graph provided $n\geq n_0(H).$ A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Note that $B_{r+1}$ is a color-critical graph with $\chi(B_{r+1})=3$. Simonovits' theorem implies that $T_{n,2}$ is the only extremal graph for $B_{r+1}$-free graphs of sufficiently large order $n$. Furthermore, Edwards and independently Khad\v{z}iivanov and Nikiforov completely confirmed Erd\H{o}s' booksize conjecture and obtained that ex$(n, B_{r+1})=e(T_{n,2})$ for $n\geq n_0(B_{r+1})=6r$. Recently, Zhai and Lin [J. Graph Theory 102 (2023) 502-520] investigated the problem of booksize from a spectral perspective. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant results, we in this paper focus on the Tur\'{a}n problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For $r = 0,$ Erd\H{o}s proved a nice result: If $G$ is a non-bipartite triangle-free graph on $n$ vertices, then $e(G)\leq\big\lfloor\frac{(n-1)^{2}}{4}\big\rfloor+1$. For general $r\geq1,$ we determine the exact value of Tur\'{a}n number of $B_{r+1}$ in non-bipartite graphs and characterize all extremal graphs provided $n$ is sufficiently large. An interesting phenomenon is that the Tur\'{a}n numbers and extremal graphs are completely different for $r=0$ and general $r\geq1.$