Flexibility of planar graphs without $C_4$ and $C_5$

Abstract

Let G be a {C 4 , C 5 }-free planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there exists an L-coloring respecting at least a constant fraction of the preferences.

Flexible list colorings in graphs with special degeneracy conditions

“…Apart from the original paper introducing flexibility [11], where some basic results in terms of maximum average degree were established, the main focus in flexibility research has been on planar graphs. In particular, for many subclasses G of planar graphs, there has been a vast effort to reduce the gap between the choosability of G and the list size needed for flexibility in G. As of now, some tight bounds on list sizes are known: namely, triangle-free 2 planar graphs [10], {C 4 , C 5 }-free planar graphs [21], and {K 4 , C 5 , C 6 , C 7 , B 5 }-free 3 planar graphs [17] are flexibly 4-choosable, and planar graphs of girth 6 [9] are flexibly 3-choosable. For other subclasses G of planar graphs, an upper bound is known for the list size k required for G to be flexibly k-choosable [7,19].…”

“…In particular, for many subclasses  of planar graphs, there has been a vast effort to reduce the gap between the choosability of  and the list size needed for flexibility in . As of now, some tight bounds on list sizes are known: namely, triangle-free 2 planar graphs [10], C C { , } 4 5 -free planar graphs [21], and…”

Flexible List Colorings in Graphs with Special Degeneracy Conditions

“…Apart from the original paper introducing flexibility [10], where some basic results in terms of maximum average degree were established, the main focus in flexibility research has been on planar graphs. In particular, for many subclasses G of planar graphs, there has been a vast effort to reduce the gap between the choosability of G and the list size needed for flexibility in G. As of now, some tight bounds on list sizes are known: namely, triangle-free planar graphs [8], {C 4 , C 5 }-free planar graphs [20], and {K 4 , C 5 , C 6 , C 7 , B 5 }-free 1 planar graphs [16] are flexibly 4-choosable, and planar graphs of girth 6 [9] are flexibly 3-choosable. For other subclasses G of planar graphs, only an upper-bound is known for the list size k required for G to be flexibly k-choosable [6,18]; see [6] for a comprehensive overview and a discussion of the related results.…”

On sufficient conditions for planar graphs to be 5-flexible

“…Their first theorem has strengthened the result of Masařík, which is a good bound up to the list size compared to choosability, since the conjecture that K − 4 -free planar graphs are 4-choosable is still open. Nowadays, Yang and the author [15] extended the third theorem of Choi et. al, they showed that every {C 4 , C 5 }-free planar graph is weighted ε-flexible with a 4-assignment, which is the best possible with respect to the list size, since Voigt [13] gave a planar graph without C 4 and C 5 is not 3-choosable.…”