Flexibility of triangle‐free planar graphs

Abstract

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle‐free and all lists have size at least four, then there exists an L‐coloring respecting at least a constant fraction of the preferences.

Flexibility of planar graphs of girth at least six

“…Let us remark that the proposed list sizes match the best possible bounds guaranteeing the existence of a coloring from the lists [7,8]. In [1], we gave a positive answer to the part (b).…”

On sufficient conditions for planar graphs to be 5-flexible

“…The following lemma derived from Choi et al provide us with a unified approach to deal with the weighted flexibility of any graph with forbidden subgraphs, which also strengthen the key lemma implicitly presented by Dvořák, Norin and Postle in [7], and explicitly formulated as Lemma 4 in [5].…”

“…In particular, there are lots of results respect to forbidding some configurations in planar graphs. Dvořák, Masařík, Musílek and Pangrác [5] proved that planar graphs without triangles are weighted ε-flexible with a 4-assignment, the result they gave is the best possible with respect to the list size since planar graphs without triangles are 4-choosable.…”

Flexibility of planar graphs without 4-cycles

“…However, (if it is true) it might be difficult to obtain such a result since even the result of Thomassen [9] for choosability is very involved. In particular, compare it to a rather easy proof [6] for choosability of triangle-free planar graphs and still the respective result for flexibility [2] was quite technical.…”

“…Dvořák, Masařík, Musílek, and Pangrác subsequently answer two such questions. In [2] they show that triangle-free planar graphs with an assignment of lists of size 4 are weighted ε-flexible. This is optimal since there are triangle-free planar graphs that are not 3-choosable [5,11].…”