Flexibility of planar graphs—Sharpening the tools to get lists of size four

Flexible list colorings in graphs with special degeneracy conditions

“…For each Δ 2 ≥ , does there exist a value ε ε = (Δ) > 0 such that any nonregular connected graph G of maximum degree Δ is ε-flexibly Δ-choosable? Later, in [7], Choi et al asked another specific question regarding degeneracy and flexibility. Knowing that outer-planar graphs are 2-degenerate, the authors asked: Question 1.2 (Choi et al [7]).…”

“…Later, in [7], Choi et al asked another specific question regarding degeneracy and flexibility. Knowing that outer-planar graphs are 2-degenerate, the authors asked: Question 1.2 (Choi et al [7]). Are outer-planar graphs flexibly 3-choosable?…”

“…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]. However, these upper bounds are not known to be tight; see [7] for a comprehensive overview and a discussion of the related results.…”

“…An interesting special case of unweighted flexibility, which was brought up recently in [7], arises when each vertex of $$G$$ has exactly one preferred color, that is, $$\text{dom}(r)=V(G)$$. We call such a request $$r$$ widespread .…”

“…As of now, some tight bounds on list sizes are known: namely, triangle‐free planar graphs [10], $$\{{C}_{4},{C}_{5}\}$$‐free planar graphs [21], and $$\{{K}_{4},{C}_{5},{C}_{6},{C}_{7},{B}_{5}\}$$‐free planar graphs [17] are flexibly 4‐choosable, and planar graphs of girth 6 [9] are flexibly 3‐choosable. For other subclasses $${\mathscr{G}}$$ of planar graphs, an upper bound is known for the list size $$k$$ required for $${\mathscr{G}}$$ to be flexibly $$k$$‐choosable [7,19]. However, these upper bounds are not known to be tight; see [7] for a comprehensive overview and a discussion of the related results.…”