List coloring with requests

Abstract

Let G be a graph with a list assignment L. Suppose a preferred color is given for some of the vertices; how many of these preferences can be respected when L-coloring G? We explore several natural questions arising in this context, and propose directions for further research.

Flexibility of planar graphs of girth at least six

“…In this note, we consider coloring from lists of size three, compare the part (c). While we do not resolve this (we suspect quite difficult) case, we give a positive answer to an easier version of this question, raised as Problem 6 in [2]: Girth at least six is sufficient to ensure weighted flexibility with lists of size three. Note this improves upon the result of Dvořák, Norin, and Postle [2] who proved that girth at least 12 is sufficient.…”

“…The following lemma is implicit in Dvořák et al [2] and appears explicitly in [1]. , and let I be the set of neighbors of v in G Z − ; since G has girth at least g and…”

“…In a proper graph coloring, we want to assign to each vertex of a graph one of a fixed number of colors in such a way that adjacent vertices receive distinct colors. Dvořák, Norin, and Postle [2] (motivated by a similar notion considered by Dvořák and Sereni [3]) introduced the following graph coloring question: If some vertices of the graph have a preferred color, is it possible to properly color the graph so that at least a constant fraction of the preferences is satisfied? As it turns out, this question is trivial in the ordinary proper coloring setting with a bounded number of colors (the answer is always positive [2]), but in the list coloring setting this gives rise to a number of interesting problems.…”

“…Dvořák, Norin, and Postle [2] established basic properties of the concept and proved that several natural graph classes are flexible: For every , there exists such that ‐degenerate graphs with assignments of lists of size are weighted ‐flexible. There exists such that every planar graph with assignment of lists of size 6 is ‐flexible. There exists such that every planar graph of girth at least five with assignment of lists of size 4 is ‐flexible. They also raised a number of interesting questions, including the following one.…”

Flexibility of triangle‐free planar graphs

“…Dvořák et al [4] asked a related question: In a given class of graphs, is it always possible to satisfy at least a constant proportion of the requests? We say that a graph with the list assignment is ‐ flexible if every request is ‐satisfiable, and it is weighted ‐ flexible if every weighted request is ‐satisfiable (of course, weighted ‐flexibility implies ‐flexibility).…”