Every triangle-free induced subgraph of the triangular lattice is $(5m,2m)$-choosable

TL;DR

This paper proves that all finite triangle-free induced subgraphs of the triangular lattice are $(5m,2m)$-choosable for any integer $m \,\geq 1$, extending understanding of list coloring in lattice graphs.

Contribution

It establishes a universal $(5m,2m)$-choosability result for triangle-free subgraphs of the triangular lattice, a significant extension in graph coloring theory.

Findings

All such subgraphs are $(5m,2m)$-choosable for any $m \,\geq 1$

The result applies to finite induced subgraphs of the triangular lattice

Provides a new bound in list coloring of lattice graphs

Abstract

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertex, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper proves that for any integer $m\ge 1$, every finite triangle-free induced subgraph of the triangular lattice is $(5m,2m)$-choosable.