Size conditions for admissible or consecutive even cycles in graphs

Abstract

In 2022, Gao, Huo, Liu, and Ma proved that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles, where a set of $k$ cycles is said to be admissible if their lengths form an arithmetic progression with common difference one or two. In this paper, we provide a sharp size analogue of their result and characterize the extremal graphs attaining the lower bound. In 2016, Verstraëte conjectured that every $n$-vertex graph $G$ containing no $k$ cycles of consecutive even lengths has at most $(2k+1)(n-1)/2$ edges, with equality only if every block of $G$ is a clique of order $2k+1$. We prove this conjecture for $2k+2\leq n\leq 4k+1$, and in fact obtain a stronger result in this range.

Theorems & Definitions (14)

• Conjecture I: Liu and Ma 12
• Theorem 1: Gao, Huo, Liu and Ma 10
• Theorem 2
• Theorem 3: Liu and Ma 12
• Conjecture II: Verstraëte V1
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 7: Block Leaf Transformation
• Lemma 9: Brandt 3