Extremal Graph for Intersecting Odd Cycles

Abstract

An extremal graph for a graph H on n vertices is a graph on n vertices with maximum number of edges that does not contain H as a subgraph. Let T n,r be the Turán graph, which is the complete r-partite graph on n vertices with part sizes that differ by at most one. The well-known Turán Theorem states that T n,r is the only extremal graph for complete graph K r+1 . Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

Extremal graphs for the k‐flower

“…where the last inequality holds since any matching in 1− that extends a matching in 1− [ ( ) ∩ 1− ] can have at most | 1− | − ( ) additional edges (even though some endpoints of additional edges may be in ( ) ∩ 1− ). This proves (9).…”

“…Moreover, denote by , 1 , 2 a -flower that contains 1 odd cycles (cycles with odd number of vertices) and 2 triangles (we always have 1 ≥ 2 , since triangle is also an odd cycle). Recently, by the stability method, Hou et al [9] determined the extremal graphs for the -flower consisting of odd cycles with the same length. For each , 1 , 2 with 1 ≥ 1, the chromatic number of , 1 , 2 is three, and so by Theorem 1.2, ( , , 1 , 2 ) = (1 + (1)) 2 ∕4.…”

“…Recently, by the stability method, Hou et al. determined the extremal graphs for the k ‐flower consisting of k odd cycles with the same length. For each with , the chromatic number of is three, and so by Theorem , .…”

Turán number and decomposition number of intersecting odd cycles

“…Lemma 2.5 (Lemma 8 in [10]). Let n 0 be an integer and let G be a graph on n n 0 + n 0 2 vertices with e(G) = e(T n,2 ) + j for some integer j 0.…”

A survey on spectral conditions for some extremal graph problems

“…When q is an odd integer, we can see that χ(C k,q ) = 3, the Erdős-Stone-Simonovits theorem also implies that ex(n, C k,q ) = n 2 /4 + o(n 2 ). In 2016, Hou, Qiu and Liu [93] determined exactly the extremal number for C k,q with k ≥ 1 and odd integer q ≥ 5.…”