Turán Number of Books in Non‐Bipartite Graphs

ABSTRACT Let be the Turán number of for a given graph . A graph is color‐critical if it contains an edge whose removal reduces its chromatic number. Simonovits' chromatic critical edge theorem states that if is color‐critical with , then there exists an such that ex and the Turán graph is the only extremal graph provided . A book graph is a set of triangles with a common edge, where is an integer. Note that is a color‐critical graph with . Simonovits' theorem implies that is the only extremal graph for ‐free graphs of sufficiently large order . Furthermore, Edwards and independently Khadžiivanov and Nikiforov completely confirmed Erdős' booksize conjecture and obtained that ex for . Recently, Zhai and Lin investigated the problem of booksize from a spectral perspective. Note that the extremal graph is bipartite. Motivated by the above elegant results, we in this paper focus on the Turán problem of non‐bipartite ‐free graphs of order . For , Erdős proved a nice result: If is a non‐bipartite triangle‐free graph on vertices, then . For general , we determine the exact value of the Turán number of in non‐bipartite graphs and characterize all extremal graphs provided is sufficiently large. An interesting phenomenon is that the Turán numbers and extremal graphs are completely different for and general .