Independent Sets in the Union of Two Hamiltonian Cycles

Ron Aharoni,Daniel Soltész

doi:10.37236/6493

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.

Highlights

In this paper we study a “pigeonhole” phenomenon for Hamiltonian cycles - in a large enough set of such cycles there are necessarily two that are close, in the sense that theirThroughout the paper, unless otherwise stated the size of the vertex set of any graph mentioned is denoted by n
We study f (n, k) in the case where k is a linear function of n, namely k = cn
This is very natural as the independence number of a Hamiltonian cycle grows roughly like a linear function of n

Summary

Introduction

In this paper we study a “pigeonhole” phenomenon for Hamiltonian cycles - in a large enough set of such cycles there are necessarily two that are close, in the sense that their. Since a two-miltonian graph satisfies ∆ 4, the following results will be useful for us: Theorem 5. We excluded K5 from the definition of two-miltonian graphs because the structure of K4s is different in K5 than in other graphs that are the union of two Hamiltonian cycles. Lemma 8 states that if G is two miltonian, we can use theorem 5 on the K4-free part of G to obtain a large independent set and we can further enlarge it by adding a vertex from each K4 maintaining independence. The authors feel that in Lemma 8 the assumption that G is two-miltonian can be replaced by different assumptions, see Remark 32 This Lemma is the core of the argument for the lower bound in Theorem 3 and in the proof of the following theorem.

A threshold phenomenon

K4-free graphs

Then each union

A lower bound on ct