Extremal graphs for the k‐flower

Abstract

The Turán number of a graph H, ex(n,H), is the maximum number of edges in any graph of order n that does not contain an H as a subgraph. A graph on 2k+1 vertices consisting of k triangles that intersect in exactly one common vertex is called a k‐fan, and a graph consisting of k cycles that intersect in exactly one common vertex is called a k‐flower. In this article, we determine the Turán number of any k‐flower containing at least one odd cycle and characterize all extremal graphs provided n is sufficiently large. Erdős, Füredi, Gould, and Gunderson determined the Turán number for the k‐fan. Our result is a generalization of their result. The addition aim of this article is to draw attention to a powerful tool, the so‐called progressive induction lemma of Simonovits.

A survey on spectral conditions for some extremal graph problems

“…In 2018, Hou, Qiu and Liu [94] and Yuan [181] independently determined the extremal number of H s,k for s ≥ 0 and k ≥ 1. Let F n,s,k be the family of graphs with each member being a Turán graph T 2 (n) with a graph H embedded in one partite set, where…”

“…Theorem 1.77 (Hou-Qiu-Liu [94]; Yuan [181]). For every graph H s,t 1 ,...,t k with s ≥ 0 and k ≥ 1, there exists n 0 such that for all n ≥ n 0 , we have…”

Extremal graphs for edge blow-up of graphs

“…Thought the proof of Lemma 2.6 in [3], it is not difficult to see that if the equality holds in (6), then ( 7) is satisfied and G[V i ] ∈ E k−1,k−1 which is not appeared in the original description of Lemma 2.6 in [3]. See Lemma 2.7 in [19].…”

“…It is a challenge of determining the exact Turán function for more non-bipartite graphs, although the Turán function of non-bipartite graphs is asymptotically determined by Erdős-Stone-Simonovits theorem. There are only few graphs whose Turán number were determined exactly, including edge-critical graphs [15] and some special graphs [3,17,19].…”

Anti-Ramsey numbers of graphs with some decomposition family sequences

“…In 1960s, Simonovits [20] introduced the so-called progressive induction which is similar to the mathematical induction and Euclidean algorithm and combined from them in a certain sense. The progressive induction method is key powerful for extremal problems of non-bipartite graphs, for example see [24].…”