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

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 with $n$ vertices and $k$ parts. Given a H $H$ let e x ( ) $ex(n,H)$ number of . Simonovits' critical theorem states that χ = + 1 $\chi (H)=k+1$ then there exists 0 ${n}_{0}(H)$ such | E $ex(n,H)=|E({T}_{n,k})|$ only extremal provided ≥ $n\ge {n}_{0}(H)$ Nikiforov proved spectral theorem. It asserts (which exponential V $|V(H)|$ s p ρ $e{x}_{sp}(n,H)=\rho ({T}_{n,k})$ where G $\rho (G)$ radius $G$ max { : ⊈ } $e{x}_{sp}(n,H)=\max \{\rho (G):|V(G)|=n\,\text{and}\,H \nsubseteq G\}$ In addition, either complete or odd cycle, linear book B r ${B}_{r}$ set $r$ triangles sharing common theta θ ${\theta }_{r}$ which consists two connected by three internally disjoint paths length one, two, Notice both are color-critical. this article, we prove 2 (G)\ge \rho ({T}_{n,2})$ > 13 $r\gt \frac{2}{13}n$ unless $G={T}_{n,2}$ Similarly, 10 \frac{n}{10}$ for 7 \frac{n}{7}$ even Our results imply in graphs graphs. result can viewed as version Erdős conjecture (1962) stating every -vertex $|E(G)|\gt |E({T}_{n,2})|$ 6 \frac{n}{6}.$ Moreover, our yields (G)\gt cycle t $t$ each ≤ $t\le This related to open question (2008) asks maximum c $c$ large enough order cn$