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arXiv · 2201.03633

Vertex-edge marking score of certain triangular lattices

Daniel Herden·Jonathan Meddaugh·Mark Sepanski·Isaac Echols·Nina Garcia-Montoya·Cordell Hammon·Guanjie Huang·Adam Kraus·Jorge Marchena Menendez·Jasmin Mohn·Rafael Morales Jiménez
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Abstract

The vertex-edge marking game is played between two players on a graph, $G=(V,E)$, with one player marking vertices and the other marking edges. The players want to minimize/maximize, respectively, the number of marked edges incident to an unmarked vertex. The vertex-edge coloring number for $G$ is the maximum score achievable with perfect play. Bre\v{s}ar et al., [4], give an upper bound of $5$ for the vertex-edge coloring number for finite planar graphs. It is not known whether the bound is tight. In this paper, in response to questions in [4], we show that the vertex-edge coloring number for the infinite regular triangularization of the plane is 4. We also give two general techniques that allow us to calculate the vertex-edge coloring number in many related triangularizations of the plane.

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Daniel Herden, Jonathan Meddaugh, Mark Sepanski, Isaac Echols, Nina Garcia-Montoya, Cordell Hammon, Guanjie Huang, Adam Kraus, Jorge Marchena Menendez, Jasmin Mohn, Rafael Morales Jiménez. 2022-01-10. Vertex-edge marking score of certain triangular lattices. https://arxiv.org/abs/2201.03633

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