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Combinatorics, Probability and Computing Combinatorics, Probability and Computing

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Cycle partitions of regular graphs

Part of: Graph theory

Published online by Cambridge University Press:  18 December 2020

Vytautas Gruslys  and
Shoham Letzter
Vytautas Gruslys
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK Email: v.gruslys@gmail.com
Shoham Letzter
Affiliation:
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK Email: v.gruslys@gmail.com
Corresponding
E-mail address:
v.gruslys@gmail.com
s.letzter@ucl.ac.uk
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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 is tight for bipartite graphs).

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Secondary: 05C38: Paths and cycles
Type
Paper
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Combinatorics, Probability and Computing , First View , pp. 1 - 24
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

Research was supported by Dr Max Rössler, by theWalter Haefner Foundation and by the ETH Zurich Foundation.

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