Turán number of books in non-bipartite graphs

Let ex (n, H) be the Turán 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 χ (H) =k+1, then there exists an n₀ (H) such that ex (n, H) =e (T₍, ₊) and the Turán graph T₍, ₊ is the only extremal graph provided n n₀ (H). A book graph Bₑ+₁ is a set of r+1 triangles with a common edge, where r0 is an integer. Note that Bₑ+₁ is a color-critical graph with χ (Bₑ+₁) =3. Simonovits' theorem implies that T₍, ₂ is the only extremal graph for Bₑ+₁-free graphs of sufficiently large order n. Furthermore, Edwards and independently Khadžiivanov and Nikiforov completely confirmed Erdős' booksize conjecture and obtained that ex (n, Bₑ+₁) =e (T₍, ₂) for n n₀ (Bₑ+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₍, ₂ is bipartite. Motivated by the above elegant results, we in this paper focus on the Turán problem of non-bipartite Bₑ+₁-free graphs of order n. For r = 0, Erdős proved a nice result: If G is a non-bipartite triangle-free graph on n vertices, then e (G) (n-1) ^24+1. For general r1, we determine the exact value of Turán number of Bₑ+₁ in non-bipartite graphs and characterize all extremal graphs provided n is sufficiently large. An interesting phenomenon is that the Turán numbers and extremal graphs are completely different for r=0 and general r1.