English

Choosability with union separation

Combinatorics 2015-12-25 v1

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 tkt \geq k, a (k,t)(k,t)-list assignment is a list assignment LL where L(v)k|L(v)| \geq k for all vertices vv and L(u)L(v)t|L(u)\cup L(v)| \geq t for all edges uvuv. A graph is (k,t)(k,t)-choosable if there is a proper coloring for every (k,t)(k,t)-list assignment. We explore this concept through examples of graphs that are not (k,t)(k,t)-choosable, demonstrating sparsity conditions that imply a graph is (k,t)(k,t)-choosable, and proving that all planar graphs are (3,11)(3,11)-choosable and (4,9)(4,9)-choosable.

Keywords

Cite

@article{arxiv.1512.07847,
  title  = {Choosability with union separation},
  author = {Mohit Kumbhat and Kevin Moss and Derrick Stolee},
  journal= {arXiv preprint arXiv:1512.07847},
  year   = {2015}
}

Comments

10 pages, 2 figures

R2 v1 2026-06-22T12:17:39.173Z