Even cycle creating paths

TL;DR

This paper investigates the maximum number of pairwise paths on n vertices that are $C_k$-creating, providing new upper bounds for even cycles by generalizing previous methods and constructing specific bipartite graphs.

Contribution

The paper extends existing bounds on $H(n,2k)$ for even cycles, offering a generalized upper bound applicable for all $k \

Findings

Established an upper bound for $H(n,2k)$ for all $k \\geq 2$.

Generalized previous methods to broader cycle lengths.

Identified special cases with denser bipartite graphs leading to tighter bounds.

Abstract

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the maximal number of pairwise $C_k$-creating paths (of arbitrary length) on $n$ vertices. The behaviour of $H(n,2k+1)$ is much better understood than the behaviour of $H(n,2k)$, the former is an exponential function of $n$ while the latter is larger than exponential, for every fixed $k$. We study $H(n,k)$ for fixed $k$ and $n$ tending to infinity. The only non trivial upper bound on $H(n,2k)$ was in the case where $k=2$ $$H(n,4)\leq n^{\left(1-\frac{1}{4} \right) n-o(n)}, $$ this was proved by Cohen, Fachini and K\"orner. In this paper, we generalize their method to prove that for every $k \geq 2$, $$H(n,2k) \leq n^{\left( 1- \frac{2}{3k^2-2k} \right)n-o(n)}. $$ Our proof uses constructions of bipartite, regular, $C_{2k}$-free graphs with many edges by Reiman, Benson, Lazebnik, Ustimenko and Woldar. For some special values of $k$ we can have slightly denser such bipartite graphs than for general $k$, this results in having better upper bounds on $H(n,2k)$ than stated above for these special values of $k$.