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arXiv · 1808.00851

Cycle partitions of regular graphs

Vytautas Gruslys·Shoham Letzter
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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).

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Vytautas Gruslys, Shoham Letzter. 2018-08-02. Cycle partitions of regular graphs. https://doi.org/10.1017/s0963548320000553

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