Extremal numbers of cycles revisited

TL;DR

This paper offers a geometric perspective on Wenger's algebraic construction, producing large graphs with no small even cycles and near-optimal edge counts, enhancing understanding of extremal cycle-free graphs.

Contribution

It provides a new geometric interpretation of Wenger's construction, improving the comprehension of extremal graphs avoiding certain cycle lengths.

Findings

Constructs large cycle-free graphs with near-maximum edges.

Provides geometric insight into algebraic graph constructions.

Achieves graphs with no cycles of length 4, 6, or 10.

Abstract

We give a simple geometric interpretation of an algebraic construction of Wenger that yields $n$-vertex graphs with no cycle of length $4$, $6$ or $10$ and close to the maximum number of edges.