On the Tur\'an number of some ordered even cycles

TL;DR

This paper extends classical extremal graph theory results to ordered graphs, establishing tight bounds on the maximum edges avoiding certain ordered cycles called bordered cycles, including a strengthened bound for 6-cycles.

Contribution

It introduces and analyzes ordered variants of cycle avoidance, providing tight bounds for bordered cycles and improving known results for specific cycle lengths.

Findings

Maximum edges avoiding bordered cycles of length at most 2k is Θ(n^{1+1/k})

Forbidding bordered orderings of 6-cycle yields an O(n^{4/3}) edge bound

Extends classical results to ordered graph settings with new tight bounds

Abstract

A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for $k=2,3,5$. In this paper we study ordered variants of this problem and prove some tight estimates for a certain class of ordered cycles that we call bordered cycles. In particular, we show that the maximum number of edges in an ordered graph avoiding bordered cycles of length at most $2k$ is $\Theta(n^{1+1/k})$. Strengthening the result of Bondy and Simonovits in the case of 6-cycles, we also show that it is enough to forbid these bordered orderings of the 6-cycle to guarantee an upper bound of $O(n^{4/3})$ on the number of edges.