NP-Completeness of Perfect Matching Index of Cubic Graphs

Authors Martin Škoviera , Peter Varša

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Author Details

Martin Škoviera
  • Department of Computer Science, Comenius University, Bratislava, Slovakia
Peter Varša
  • Department of Computer Science, Comenius University, Bratislava, Slovakia


The authors wish to thank Edita Máčajová for helpful discusions.

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph coloring
  • cubic graph
  • edge colouring
  • snark
  • perfect matching
  • covering
  • NP-completeness


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