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DOI: 10.4230/LIPIcs.STACS.2022.56
NP-Completeness of Perfect Matching Index of Cubic Graphs

Authors: Martin Škoviera and Peter Varša

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
The perfect matching index of a cubic graph G, denoted by π(G), is the smallest number of perfect matchings needed to cover all the edges of G; it is correctly defined for every bridgeless cubic graph. The value of π(G) is always at least 3, and if G has no 3-edge-colouring, then π(G) ≥ 4. On the other hand, a long-standing conjecture of Berge suggests that π(G) never exceeds 5. It was proved by Esperet and Mazzuoccolo [J. Graph Theory 77 (2014), 144-157] that it is NP-complete to decide for a 2-connected cubic graph whether π(G) ≤ 4. A disadvantage of the proof (noted by the authors) is that the constructed graphs have 2-cuts. We show that small cuts can be avoided and that the problem remains NP-complete even for nontrivial snarks - cyclically 4-edge-connected cubic graphs of girth at least 5 with no 3-edge-colouring. Our proof significantly differs from the one due to Esperet and Mazzuoccolo in that it combines nowhere-zero flow methods with elements of projective geometry, without referring to perfect matchings explicitly.
Cite as

Martin Škoviera and Peter Varša. NP-Completeness of Perfect Matching Index of Cubic Graphs. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{skoviera_et_al:LIPIcs.STACS.2022.56,
  author =	{\v{S}koviera, Martin and Var\v{s}a, Peter},
  title =	{{NP-Completeness of Perfect Matching Index of Cubic Graphs}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{56:1--56:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.56},
  URN =		{urn:nbn:de:0030-drops-158667},
  doi =		{10.4230/LIPIcs.STACS.2022.56},
  annote =	{Keywords: cubic graph, edge colouring, snark, perfect matching, covering, NP-completeness}
}
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