Bipancyclic subgraphs in random bipartite graphs

TL;DR

This paper proves that in random bipartite graphs with sufficient edge probability, large Hamiltonian subgraphs almost surely contain cycles of all even lengths, extending classical bipancyclicity results.

Contribution

It establishes a tight resilience property for bipancyclicity in random bipartite graphs, extending classical theorems to probabilistic settings.

Findings

For p(n) >> n^{-2/3}, G(n,n,p) is almost surely bipancyclic.

Any Hamiltonian subgraph with more than half the edges is bipancyclic.

The result is tight in terms of p(n) range and edge proportion.

Abstract

A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph $G(n,n,p)$ with $p(n)\gg n^{-2/3}$ asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph $G'$ of $G(n,n,p)$ with more than $(1/2+o(1))n^2p$ edges is bipancyclic. This result is tight in two ways. First, the range of $p$ is essentially best possible. Second, the proportion 1/2 of edges cannot be reduced. Our result extends a classical theorem of Mitchem and Schmeichel.