Vertex-edge marking score of certain triangular lattices

TL;DR

This paper determines the vertex-edge coloring number for the infinite regular triangular lattice as 4, providing new techniques to analyze similar planar graphs and addressing an open question in graph theory.

Contribution

It establishes the vertex-edge coloring number as 4 for the infinite triangular lattice and introduces methods applicable to other planar triangularizations.

Findings

Vertex-edge coloring number for infinite triangular lattice is 4

Developed two techniques for analyzing related triangularizations

Addressed an open question on the tightness of the bound

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.