Cycle partitions of regular graphs

TL;DR

This paper proves a conjecture about partitioning the vertices of regular graphs into paths or cycles when the degree is proportional to the number of vertices, extending previous results and providing tight bounds for bipartite graphs.

Contribution

It confirms Magnant and Martin's conjecture for large degree regular graphs and introduces a method to partition such graphs into cycles, also establishing tight bounds for bipartite cases.

Findings

Partition of regular graphs into paths when degree is proportional to vertices

Partition of bipartite regular graphs into paths with tight bounds

Improved results over previous work for large degree regular graphs

Abstract

Magnant and Martin conjectured that the vertex set of any $d$-regular graph $G$ on $n$ vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$, improving a result of Han, who showed that in this range almost all vertices of $G$ can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact, our proof gives a partition of $V(G)$ into cycles. We also show that, if $d = \Omega(n)$ and $G$ is bipartite, then $V(G)$ can be partitioned into $n / (2d)$ paths (this bound in tight for bipartite graphs).