Flexible List Colorings in Graphs with Special Degeneracy Conditions

Abstract

For a given ε > 0, we say that a graph G is ε-flexibly k-choosable if the following holds: for any assignment L of lists of size k on V (G), if a preferred color is requested at any set R of vertices, then at least ε|R| of these requests may be satisfied by some L-coloring. We consider flexible list colorings in several graph classes with certain special degeneracy conditions. We characterize the graphs of maximum degree ∆ that are ε-flexibly ∆-choosable for some ε = ε(∆) > 0, which answers a question of Dvořák, Norin, and Postle [List coloring with requests, JGT 2019]. We also show that graphs of treewidth 2 are 1 3 -flexibly 3-choosable, answering a question of I. Choi et al. [arXiv 2020], and we give conditions for list assignments by which graphs of treewidth k are 1 k+1 -flexibly (k + 1)-choosable. We show furthermore that graphs of treedepth k are 1 k -flexibly k-choosable. Finally, we introduce a notion of flexible degeneracy, which strengthens flexible choosability, and we show that apart from a well-understood class of exceptions, three-connected non-regular graphs of maximum degree ∆ are flexibly (∆ − 1)-degenerate.

Robust Connectivity of Graphs on Surfaces

“…In our proof of the theorem, we use a greedy procedure in which we iteratively remove from R a vertex with neighbors in the greatest number of distinct components of G \ R. When the greedy procedure terminates, only one component of G \ R remains, and the remaining vertices in R form the set R ′ . A weaker version of the theorem was shown in [8] (Lemma 5.5) using a crude analysis of the same greedy method, but here we give a more careful analysis in order to obtain a better lower bound for the size of R ′ .…”

“…taken over all spanning trees T of G, in which case κ ρ (G) = min R⊆V (G) R =∅ ℓ(G, R). We may think of robust connectivity in terms of a one-turn game in which the first player chooses a set R of vertices in G, and then the second player attempts to find a spanning tree in G using as many vertices of R as leaves as possible; see [8] for details. This one-turn can be also compared with the one-turn matching game used by Matuschke, Skutella, and Soto to define robust matchings [31].…”

“…This parameter was formerly called game connectivity in[9] and older versions of[8]. However, we believe that the term "robust connectivity" better suits the properties of this parameter.2 This theorem appears as Theorem 5.6 in the full version on arXiv[8].…”