On Weak Flexibility in Planar Graphs

Abstract

Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex v in some subset of V (G) has a request for a certain color r(v) in its list of colors L(v). The goal is to find an L coloring satisfying many, but not necessarily all, of the requests.The main studied question is whether there exists a universal constant ε > 0 such that any graph G in some graph class C satisfies at least ε proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G ∈ C on vertex set V , with any list assignment L of size k for each vertex, and for every R ⊆ V and a request vector (r(v) : v ∈ R, r(v) ∈ L(v)), there exists an L-coloring of G satisfying at least ε|R| requests. If this is true, then C is called ε-flexible for lists of size k.Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where R = V . We further develop this direction by introducing a tool to handle weak flexibility. We demonstrate this new tool by showing that for every positive integer b there exists ε(b) > 0 so that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B b is weakly ε(b)flexible for lists of size 4 (here K n , C n and B n are the complete graph, a cycle, and a book on n vertices, respectively). We also show that the class of planar graphs without K 4 , C 5 , C 6 , C 7 , B 5 is ε-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.

Flexible list colorings: Maximizing the number of requests satisfied

“…Several papers [13,10,22,27,11,28] have explored (k, ǫ)-flexibility, with k ∈ {5, 4, 3}, of planar graphs with large enough girth or excluding certain cycles. A 'weak' notion of ǫ-flexibility where the domain of the request set is always the entire vertex set is studied in the context of planar graphs in [8,21].…”

Flexible list colorings: Maximizing the number of requests satisfied

“…It has been shown that there exists an ϵ > 0 such that every planar graph G is (6, ϵ)-flexible [14]. Several papers [9,11,12,14,22,23,28,29] have explored k ( , ϵ)-flexibility, with k {5, 4, 3} ∈ , of planar graphs with large enough girth or excluding certain cycles. A "weak" notion of ϵ-flexibility where the domain of the request set is always the entire vertex set is studied in the context of planar graphs in [9,22].…”

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].…”

“…5 -free 3 planar graphs [17] are flexibly 4-choosable, and planar graphs of girth 6 [9] are flexibly 3-choosable. For other subclasses  of planar graphs, an upper bound is known for the list size k required for  to be flexibly k-choosable [7,19].…”