A strengthening of the spectral chromatic critical edge theorem: Books and theta graphs

• Published in: Journal of graph theory 2023-03, Vol.102 (3), p.502-520
• Main Authors: Zhai, Mingqing, Lin, Huiqiu
• Format: Article
• Language: English
• Subjects: Apexes; chromatic critical edge theorem; Color; consecutive cycles; Graph theory; Graphs; spectral extrema; Theorems; theta graph
• ISSN: 0364-9024; 1097-0118
• DOI: 10.1002/jgt.22883

A graph is color‐critical if it contains an edge whose removal reduces its chromatic number. Let T n , k ${T}_{n,k}$ be the Turán graph with n $n$ vertices and k $k$ parts. Given a graph H $H$, let e x ( n , H ) $ex(n,H)$ be the Turán number of H $H$. Simonovits' chromatic critical edge theorem states that if H $H$ is color‐critical with χ ( H ) = k + 1 $\chi (H)=k+1$, then there exists an n 0 ( H ) ${n}_{0}(H)$ such that e x ( n , H ) = | E ( T n , k ) | $ex(n,H)=|E({T}_{n,k})|$ and the Turán graph T n , k ${T}_{n,k}$ is the only extremal graph provided n ≥ n 0 ( H ) $n\ge {n}_{0}(H)$. Nikiforov proved a spectral chromatic critical edge theorem. It asserts that if H $H$ is color‐critical and χ ( H ) = k + 1 $\chi (H)=k+1$, then there exists an n 0 ( H ) ${n}_{0}(H)$ (which is exponential with | V ( H ) | $|V(H)|$) such that e x s p ( n , H ) = ρ ( T n , k ) $e{x}_{sp}(n,H)=\rho ({T}_{n,k})$ and T n , k ${T}_{n,k}$ is the only extremal graph provided n ≥ n 0 ( H ) $n\ge {n}_{0}(H)$, where ρ ( G ) $\rho (G)$ is the spectral radius of G $G$ and e x s p ( n , H ) = max { ρ ( G ) : | V ( G ) | = n and H ⊈ G } $e{x}_{sp}(n,H)=\max \{\rho (G):|V(G)|=n\,\text{and}\,H \nsubseteq G\}$. In addition, if H $H$ is either a complete graph or an odd cycle, then n 0 ( H ) ${n}_{0}(H)$ is linear with | V ( H ) | $|V(H)|$. A book graph B r ${B}_{r}$ is a set of r $r$ triangles sharing a common edge and a theta graph θ r ${\theta }_{r}$ is a graph which consists of two vertices connected by three internally disjoint paths with length one, two, and r $r$. Notice that both B r ${B}_{r}$ and θ r ${\theta }_{r}$ are color‐critical. In this article, we prove that if ρ ( G ) ≥ ρ ( T n , 2 ) $\rho (G)\ge \rho ({T}_{n,2})$, then G $G$ contains a book B r ${B}_{r}$ with r > 2 13 n $r\gt \frac{2}{13}n$ unless G = T n , 2 $G={T}_{n,2}$. Similarly, we prove that if ρ ( G ) ≥ ρ ( T n , 2 ) $\rho (G)\ge \rho ({T}_{n,2})$, then G $G$ contains a theta graph θ r ${\theta }_{r}$ with r > n 10 $r\gt \frac{n}{10}$ for odd r $r$ and r > n 7 $r\gt \frac{n}{7}$ for even r $r$ unless G = T n , 2 $G={T}_{n,2}$. Our results imply that n 0 ( H ) ${n}_{0}(H)$ in the spectral chromatic critical edge theorem is linear with | V ( H ) | $|V(H)|$ for book graphs and theta graphs. Our result for book graphs can be viewed as a spectral version of an Erdős conjecture (1962) stating that every n $n$‐vertex graph with | E ( G ) | > | E ( T n , 2 ) | $|E(G)|\gt |E({T}_{n,2})|$ contains a book graph B r ${B}_{r}$