# New bounds on even cycle creating Hamiltonian paths using expander graphs

## TL;DR

This paper refines bounds on the maximum number of Hamiltonian paths that create a specific even cycle in the union of two graphs, using expander graph properties to close previous exponential gaps.

## Contribution

It provides tighter bounds on the number of Hamiltonian paths creating even cycles, especially for $C_4$, and improves bounds for larger even cycles, advancing understanding of graph union properties.

## Key findings

- Established near-tight bounds for $H_n(C_4)$, closing the exponential gap.
- Improved upper bounds on $H_n(C_{2k})$ for larger even cycles.
- Enhanced bounds on pairwise reversing permutations.

## Abstract

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved \[n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.\] In this paper we close the superexponential gap between their lower and upper bounds by proving \[n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.\] We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, and we present a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.