Independent sets in the union of two Hamiltonian cycles

TL;DR

This paper studies the maximum number of Hamiltonian cycles on an n-element set that do not share large independent sets, revealing a threshold phenomenon based on a linear function of n, with bounds on the critical constant.

Contribution

It introduces a threshold phenomenon for the maximum number of Hamiltonian cycles with limited shared independent sets, providing bounds on the critical constant and a technical lemma relating independence number and K4 subgraphs.

Findings

Existence of a threshold constant c_t between 0.26 and 0.36.

For c < c_t, the maximum number is bounded by a constant.

For c > c_t, the maximum number grows exponentially with n.

Abstract

Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)$ is exponentially large in $n ~(n \to \infty)$. We prove that $0.26 < c_t < 0.36$, but the exact value of $c_t$ is not determined. For the lower bound we prove a technical lemma, which for graphs that are the union of two Hamiltonian cycles establishes a relation between the independence number and the number of $K_4$ subgraphs. A corollary of this lemma is that if a graph $G$ on $n>12$ vertices is the union of two Hamiltonian cycles and $\alpha(G)=n/4$, then $V(G)$ can be covered by vertex-disjoint $K_4$ subgraphs.