A Note on Odd Colorings of 1-Planar Graphs
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Publication:6390229
DOI10.1016/J.DAM.2023.01.011arXiv2202.02586MaRDI QIDQ6390229
Daniel W. Cranston, Michael Lafferty, Z. X. Song
Publication date: 5 February 2022
Abstract: A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petruv{s}evski and v{S}krekovski, who proved that every planar graph admits an odd -coloring; they also conjectured that every planar graph admits an odd -coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petruv{s}evski, and v{S}krekovski. Building on the work of Caro, Petruv{s}evski, and v{S}krekovski, Petr and Portier then further proved that every planar graph admits an odd -coloring. In this note we prove that every 1-planar graph admits an odd -coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge.
Planar graphs; geometric and topological aspects of graph theory (05C10) Coloring of graphs and hypergraphs (05C15)
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