On Weak Flexibility in Planar Graphs

Abstract

Recently, Dvořák, Norin, and Postle introduced flexibility as an extension of list coloring on graphs (J Graph Theory 92(3):191–206, 2019, https://doi.org/10.1002/jgt.22447). 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 varepsilon >0 such that any graph G in some graph class mathscr {C} satisfies at least varepsilon proportion of the requests. More formally, for k > 0 the goal is to prove that for any graph G in mathscr {C} on vertex set V, with any list assignment L of size k for each vertex, and for every R subseteq V and a request vector (r(v): vin R, ~r(v) in L(v)), there exists an L-coloring of G satisfying at least varepsilon |R| requests. If this is true, then mathscr {C} is called varepsilon-flexible for lists of size k. Choi, Clemen, Ferrara, Horn, Ma, and Masařík (Discrete Appl Math 306:20–132, 2022, https://doi.org/10.1016/j.dam.2021.09.021) 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 varepsilon (b)>0 so that the class of planar graphs without K_4, C_5 , C_6 , C_7, B_b is weakly varepsilon (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 varepsilon-flexible for lists of size 4. The results are tight as these graph classes are not even 3-colorable.