Choosability with union separation

TL;DR

This paper introduces and studies a new graph coloring refinement called choosability with union separation, proving results for planar graphs and exploring conditions for graph choosability based on list union sizes.

Contribution

It defines the concept of choosability with union separation, establishes key properties, and proves that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.

Findings

All planar graphs are $(3,11)$-choosable.

All planar graphs are $(4,9)$-choosable.

Sparsity conditions imply $(k,t)$-choosability.

Abstract

List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t \geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| \geq k$ for all vertices $v$ and $|L(u)\cup L(v)| \geq t$ for all edges $uv$. A graph is $(k,t)$-choosable if there is a proper coloring for every $(k,t)$-list assignment. We explore this concept through examples of graphs that are not $(k,t)$-choosable, demonstrating sparsity conditions that imply a graph is $(k,t)$-choosable, and proving that all planar graphs are $(3,11)$-choosable and $(4,9)$-choosable.