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arXiv · 2205.09317

Odd coloring of 2-boundary planar graphs and beyond

Weichan Liu·Mengke Qi·Xin Zhang
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Abstract

In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. 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. Petru\v{s}evski and \v{S}krekovski conjectured in 2022 that every planar graph admits an odd 5-coloring. We confirm this conjecture for 2-boundary planar graphs. Moreover, we present several questions regarding 2-boundary planar graphs that are of independent interest.

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Weichan Liu, Mengke Qi, Xin Zhang. 2022-05-19. Odd coloring of 2-boundary planar graphs and beyond. https://arxiv.org/abs/2205.09317

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