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

Abstract

AbstractA graph is color‐critical if it contains an edge whose removal reduces its chromatic number. Let be the Turán graph with vertices and parts. Given a graph , let be the Turán number of . Simonovits' chromatic critical edge theorem states that if is color‐critical with , then there exists an such that and the Turán graph is the only extremal graph provided . Nikiforov proved a spectral chromatic critical edge theorem. It asserts that if is color‐critical and , then there exists an (which is exponential with ) such that and is the only extremal graph provided , where is the spectral radius of and . In addition, if is either a complete graph or an odd cycle, then is linear with . A book graph is a set of triangles sharing a common edge and a theta graph is a graph which consists of two vertices connected by three internally disjoint paths with length one, two, and . Notice that both and are color‐critical. In this article, we prove that if , then contains a book with unless . Similarly, we prove that if , then contains a theta graph with for odd and for even unless . Our results imply that in the spectral chromatic critical edge theorem is linear with for book graphs and theta graphs. Our result for book graphs can be viewed as a spectral version of an Erdős conjecture (1962) stating that every ‐vertex graph with contains a book graph with Moreover, our result for theta graphs yields that every graph with contains a cycle of length for each . This is related to an open question by Nikiforov (2008) which asks for the maximum such that every graph of large enough order with contains a cycle of length for every .