License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2022.56
URN: urn:nbn:de:0030-drops-158667
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15866/
Škoviera, Martin ;
Varša, Peter
NP-Completeness of Perfect Matching Index of Cubic Graphs
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.
BibTeX - Entry
@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.dagstuhl.de/opus/volltexte/2022/15866},
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}
}
Keywords: |
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cubic graph, edge colouring, snark, perfect matching, covering, NP-completeness |
Collection: |
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39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) |
Issue Date: |
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2022 |
Date of publication: |
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09.03.2022 |