Non-bipartite graphs without theta subgraphs

Fix a color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Simonovits' chromatic critical edge theorem and Nikiforov's spectral chromatic critical edge theorem imply that $T_{n,r}$ is the extremal graph with the maximum size and the maximum spectral radius over all $H$-free graphs of order $n$, respectively. Since $T_{n,r}$ is $r$-partite, it is interesting to study the Tur\'{a}n number and the spectral Tur\'{a}n number of a color-critical graph $H$ in non-$r$-partite graphs. Denote by ${\rm EX}_{r+1}(n,H)$ (resp. ${\rm SPEX}_{r+1}(n,H)$) the family of $n$-vertex $H$-free non-$r$-partite graphs with the maximum size (resp. spectral radius). Brouwer showed that any graph in $\mathrm{EX}_{r+1}(n,K_{r+1})$ is of size $e(T_{n,r})-\lfloor\frac{n}{r}\rfloor+1$ for $n\geq 2r+1$. Lin, Ning and Wu [Combin. Probab. Comput. 30 (2) (2021) 258--270], and Li and Peng [SIAM J. Discrete Math. 37 (2023) 2462--2485] characterized the unique graph in $\mathrm{SPEX}_{r+1}(n,K_{r+1})$ for $r\geq 2$. Particularly, the unique graph is of size $e(T_{n,r})-\lfloor\frac{n}{r}\rfloor+1$. Thus $\mathrm{SPEX}_{r+1}(n,K_{r+1})\subseteq \mathrm{EX}_{r+1}(n,K_{r+1})$. It is natural to conjecture that ${\rm SPEX}_{r+1}(n,H)\subseteq {\rm EX}_{r+1}(n,H)$ for arbitrary color-critical graph $H$ with $\chi(H)=r+1\geq 3$. Fix $q,r\geq 2$ with even $q$, a theta graph $\theta(1,q,r)$ is obtained from internally disjoint paths of lengths $1,q,r$, respectively by sharing a common pair of endpoints. In this paper, we prove that $\mathrm{SPEX}_{3}(n,\theta(1,q,r))\subseteq \mathrm{EX}_{3}(n,\theta(1,q,r))$ for sufficiently large $n$. Furthermore, we determine all the graphs in $\mathrm{SPEX}_{3}(n,\theta(1,q,r))$ and $\mathrm{EX}_{3}(n,\theta(1,q,r))$, respectively.