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Coverability of Graphs by Parity Regular Subgraphs

Author

Listed:
  • Mirko Petruševski

    (Faculty of Mechanical Engineering, Ss. Cyril and Methodius University, 1000 Skopje, North Macedonia
    These authors contributed equally to this work.)

  • Riste Škrekovski

    (Faculty for Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
    Faculty of Information Studies, University of Ljubljana, 8000 Novo Mesto, Slovenia
    These authors contributed equally to this work.)

Abstract

A graph is even (resp. odd) if all its vertex degrees are even (resp. odd). We consider edge coverings by prescribed number of even and/or odd subgraphs. In view of the 8-Flow Theorem, a graph admits a covering by three even subgraphs if and only if it is bridgeless. Coverability by three odd subgraphs has been characterized recently [Petruševski, M.; Škrekovski, R. Coverability of graph by three odd subgraphs. J. Graph Theory 2019 , 92 , 304–321]. It is not hard to argue that every acyclic graph can be decomposed into two odd subgraphs, which implies that every graph admits a decomposition into two odd subgraphs and one even subgraph. Here, we prove that every 3-edge-connected graph is coverable by two even subgraphs and one odd subgraph. The result is sharp in terms of edge-connectivity. We also discuss coverability by more than three parity regular subgraphs, and prove that it can be efficiently decided whether a given instance of such covering exists. Moreover, we deduce here a polynomial time algorithm which determines whether a given set of edges extends to an odd subgraph. Finally, we share some thoughts on coverability by two subgraphs and conclude with two conjectures.

Suggested Citation

  • Mirko Petruševski & Riste Škrekovski, 2021. "Coverability of Graphs by Parity Regular Subgraphs," Mathematics, MDPI, vol. 9(2), pages 1-15, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:2:p:182-:d:482145
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