Non-bipartite graphs without theta subgraphs

Fix a color-critical graph H with χ(H)=r+1≥ 3. Simonovits' chromatic critical edge theorem and Nikiforov's spectral chromatic critical edge theorem imply that Tn,r is the extremal graph with the maximum size and the maximum spectral radius over all H-free graphs of order n, respectively. Since Tn,r is r-partite, it is interesting to study the Turán number and the spectral Turán number of a color-critical graph H in non-r-partite graphs. Denote by EXᵣ₊₁(n,H) (resp. SPEXᵣ₊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 EXᵣ₊₁(n,Kᵣ₊₁) is of size e(Tn,r)-/r+1 for n≥ 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 SPEXᵣ₊₁(n,Kᵣ₊₁) for r≥ 2. Particularly, the unique graph is of size e(Tn,r)-/r+1. Thus SPEXᵣ₊₁(n,Kᵣ₊₁)⊆ EXᵣ₊₁(n,Kᵣ₊₁). It is natural to conjecture that SPEXᵣ₊₁(n,H)⊆ EXᵣ₊₁(n,H) for arbitrary color-critical graph H with χ(H)=r+1≥ 3. Fix q,r≥ 2 with even q, a theta graph $θ(1,q,r)$ 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 SPEX₃(n,θ(1,q,r))⊆ EX₃(n,θ(1,q,r)) for sufficiently large n. Furthermore, we determine all the graphs in SPEX₃(n,θ(1,q,r)) and EX₃(n,θ(1,q,r)), respectively.