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Proper conflict-free 6-coloring of planar graphs without short cycles

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  • Wang, Yunlong
  • Wang, Weifan
  • Liu, Runrun

Abstract

A proper conflict-free l-coloring of a graph G is a proper l-coloring satisfying that for any non-isolated vertex v∈V(G), there exists a color appearing exactly once in NG(v). The proper conflict-free chromatic number, denoted by χpcf(G), is the minimal integer l so that G admits a proper conflict-free l-coloring. This notion was proposed by Fabrici et al. in 2022. They focus mainly on proper conflict-free coloring of outerplanar graphs and planar graphs. They constructed a planar graph that has no proper conflict-free 5-coloring and conjectured every planar graph G has χpcf(G)≤6. In this paper, we confirm this conjecture for planar graphs without cycles of lengths 3, 5 or 6.

Suggested Citation

  • Wang, Yunlong & Wang, Weifan & Liu, Runrun, 2025. "Proper conflict-free 6-coloring of planar graphs without short cycles," Applied Mathematics and Computation, Elsevier, vol. 499(C).
    • Handle: RePEc:eee:apmaco:v:499:y:2025:i:c:s0096300325001328
    DOI: 10.1016/j.amc.2025.129405
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