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Remarks on the Local Irregularity Conjecture

Author

Listed:
  • Jelena Sedlar

    (Faculty of Civil Engineering, Architecture and Geodesy, University of Split, 21000 Split, Croatia
    Faculty of Information Studies, University of Ljubljana, 8000 Novo Mesto, Slovenia
    These authors contributed equally to this work.)

  • Riste Škrekovski

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

Abstract

A locally irregular graph is a graph in which the end vertices of every edge have distinct degrees. A locally irregular edge coloring of a graph G is any edge coloring of G such that each of the colors induces a locally irregular subgraph of G . A graph G is colorable if it allows a locally irregular edge coloring. The locally irregular chromatic index of a colorable graph G , denoted by χ irr ′ ( G ) , is the smallest number of colors used by a locally irregular edge coloring of G . The local irregularity conjecture claims that all graphs, except odd-length paths, odd-length cycles and a certain class of cacti are colorable by three colors. As the conjecture is valid for graphs with a large minimum degree and all non-colorable graphs are vertex disjoint cacti, we study rather sparse graphs. In this paper, we give a cactus graph B which contradicts this conjecture, i.e., χ irr ′ ( B ) = 4 . Nevertheless, we show that the conjecture holds for unicyclic graphs and cacti with vertex disjoint cycles.

Suggested Citation

  • Jelena Sedlar & Riste Škrekovski, 2021. "Remarks on the Local Irregularity Conjecture," Mathematics, MDPI, vol. 9(24), pages 1-10, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3209-:d:700489
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    Citations

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    Cited by:

    1. Hui Lei & Xiaopan Lian & Yongtang Shi & Ran Zhao, 2022. "Graph Classes with Locally Irregular Chromatic Index at most 4," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 903-918, December.
    2. Przybyło, Jakub, 2024. "A note on decomposing graphs to locally almost irregular subgraphs," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    3. Grzelec, Igor & Woźniak, Mariusz, 2023. "On decomposing multigraphs into locally irregular submultigraphs," Applied Mathematics and Computation, Elsevier, vol. 452(C).

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