Extended core and choosability of a graph

TL;DR

This paper establishes a link between a graph's $(a,b)$-choosability and that of its extended core, enabling new results on coloring properties of triangle-free subgraphs of the triangular lattice.

Contribution

It introduces the concept of the extended core and proves its equivalence to the original graph's $(a,b)$-choosability, facilitating new coloring results.

Findings

Proves the equivalence between $(a,b)$-choosability of a graph and its extended core.

Demonstrates $(5,2)$-choosability of triangle-free subgraphs of the triangular lattice.

Shows $(7,3)$-colorability of these subgraphs.

Abstract

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the $(a,b)$-choosability of a graph and the $(a,b)$-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove the $(5,2)$-choosability and $(7,3)$-colorability of triangle-free induced subgraphs of the triangular lattice.