Flexibility of Planar Graphs -- Sharpening the Tools to Get Lists of Size Four

A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors to each vertex guarantees an $L$-coloring. Given a list assignment $L$, an $L$-request for a vertex $v$ is a color $c\in L(v)$. In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvo\v{r}\'ak, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy"enough", as opposed to all, of the requested set of precolors. A graph $G$ is $\varepsilon$-flexible for list size $k$ if for any $k$-list assignment $L$, and any set $S$ of $L$-requests, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. It is conjectured that planar graphs are $\varepsilon$-flexible for list size $5$, yet it is proved only for list size $6$ and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masa\v{r}\'ik and show that planar graphs without $K_4^-$ are $\varepsilon$-flexible for list size $5$. We also prove that planar graphs without $4$-cycles and $3$-cycle distance at least 2 are $\varepsilon$-flexible for list size $4$. Finally, we introduce a new (slightly weaker) form of $\varepsilon$-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are $\varepsilon$-flexible for list size $5$.


Introduction
A proper (vertex) coloring of a graph G is an assignment of colors to the vertices of G such that adjacent vertices receive distinct colors. A widely studied class of problems in numerous branches of chromatic graph theory is the family of precoloring extension problems, wherein the goal is to extend some partial coloring of the graph to a coloring with a desired property. This general notion was introduced in [6,19,20], and has been studied in a breadth of settings (see, for instance, [1,2,3,4,5,8,16,24,28]). Graph coloring with preferences have many application in various fields of computer science, such as scheduling [17], register allocation [7], resource management [9] and many others.
Motivated by the work of Dvořák and Sereni in [15], Dvořák, Norin, and Postle [14] introduced the related concept of flexibility. The question of interest is the following: if some vertices of the graph have a preferred color, then is it possible to properly color the graph so that at least a constant fraction of the preferences are satisfied? While this is not a precoloring extension problem in the classical sense, the idea of retaining a set of preferred, as opposed to prescribed, colors establishes a clear link between these problems. This question is trivial in the usual proper coloring setting with a fixed number of colors, as we can permute the colors in a proper k-coloring of a graph in order to satisfy at least a 1/k-fraction of the requests [14]. On the other hand, in the setting of list coloring, the concept of flexibility gives rise to a number of interesting problems.
Before continuing, we present some formalities necessary for the results that follow. A list assignment L for a graph G is a function that assigns a set L(v) of colors to each vertex v ∈ V(G), and an L-coloring is a proper coloring ϕ such that ϕ(v) ∈ L(v) for all v ∈ V(G). A graph G is k-choosable if G is L-colorable from every list assignment L where each vertex receives at least k colors. The choosability of a graph G is the minimum k such that G is k-choosable. A weighted request is a function w that assigns a non-negative real number to each pair (v, c) where v ∈ V(G) and c ∈ L(v). Let w(G, L) = v∈V(G),c∈L (v) w(v, c). For ε > 0, we say that w is ε-satisfiable if there exists an L-coloring ϕ of G such that v∈V(G) w(v, ϕ(v)) ≥ ε · w(G, L).
An important special case is when at most one color can be requested at each vertex and all such colors have the same weight. A request for a graph G with list assignment L is a function r with dom(r) ⊆ V(G) such that r(v) ∈ L(v) for all v ∈ dom(r). If each vertex requests exactly one color, i.e., dom(r) = V(G), then such a request is widespread. For ε > 0, a request r is ε-satisfiable if there exists an L-coloring ϕ of G such that at least ε| dom(r)| vertices v in dom(r) receive color r(v). In particular, a request r is 1-satisfiable if and only if the precoloring given by r extends to an L-coloring of G.
We say that a graph G with list assignment L is ε-flexible, weakly ε-flexible, and weighted ε-flexible if every request, widespread request, and weighted request, respectively, is ε-satisfiable. 1 If G is (weighted/weakly) ε-flexible for every list assignment with lists of size k, then G is (weighted/weakly) ε-flexible for lists of size k.
The main meta-question is whether there exists a universal constant ε > 0 such that all k-choosable graphs are (weighted) ε-flexible for lists of size k. In the paper that introduced flexibility, Dvořák, Norin, and Postle [14] established some basic properties and proved several theorems in terms of degeneracy and maximum average degree. A graph G is d-degenerate if every subgraph has a vertex of degree at most d, and the degeneracy of G is the minimum d such that G is d-degenerate.
Compare this result with the corresponding trivial greedy bound for choosability; namely, d-degenerate graphs are d + 1 choosable. Since planar graphs are 5-degenerate, Theorem 1 implies that there exists ε > 0 such that every planar graph is weighted ε-flexible with lists of size 7. In the same paper [14], two bounds in terms of the maximum average degree were developed, one of which implies that there exists ε > 0 such that every planar graph is ε-flexible with lists of size 6. Since planar graphs are 5-choosable [26] and there exists a planar graph that is not 4-choosable [29], the following question is natural: Question 2. Does there exist ε > 0 such that every planar graph is (weighted) ε-flexible for lists of size 5?
Dvořák, Masařík, Musílek, and Pangrác [13] showed that planar graphs of girth at least 4 are weighted ε-flexible for lists of size 4. This is tight as planar graphs of girth at least 4 are 3-degenerate, and hence 4-choosable, and there exists a planar graph of girth 4 that is not 3-choosable [18,30]. They also showed that planar graphs of girth at least 6 are weighted ε-flexible for lists of size 3 [12]. There is still a gap, in terms of girth constraints, since planar graphs of girth at least 5 are 3-choosable [27].
Masařík made further progress towards Question 2 by showing that planar graphs without 2 4-cycles are weighted ε-flexible for lists of size 5 [23]. This raises the natural corresponding question for planar graphs without 4-cycles, which are known to be 4-choosable [22]. Question 3. Does there exist ε > 0 such that every planar graph without C 4 is (weighted) ε-flexible for lists of size 4?

Our Results
We prove a strengthening of the result in [23] towards solving Question 2. Wang and Lih [32] conjectured that planar graphs without K − 4 (K 4 without an edge, also known as a diamond) are 4-choosable. This conjecture remains open. Improving the result in [23], we prove that planar graphs without K − 4 are weighted ε-flexible for lists of size 5. This is the largest subclass of planar graphs that is known to be weighted ε-flexible for lists of size 5.
Theorem 4. There exists ε > 0 such that every planar graph without K − 4 is weighted ε-flexible for lists of size 5.
We also investigate Question 3; ε-flexibility for two subclasses of planar graphs without 4-cycles, and show that lists of size 4 are actually sufficient.
Theorem 5. There exists ε > 0 such that every planar graph without C 4 and with C 3 distance at least 2 is weighted ε-flexible for lists of size 4. Theorem 6. There exists ε > 0 such that every planar graph without C 4 , C 5 , C 6 is weighted ε-flexible for lists of size 4.
In order to prove the Theorems 4, 5, and 6, we strengthened the main tool from [14], which was explicitly presented in [13]. This tool, Lemma 13 in Section 2, allows us to construct more fine-tuned reducible configurations, thereby reducing the complexity of the discharging argument.
The concept of weak flexibility enables stronger reducible configurations when widespread requests are considered. To identify such configurations, we adapted Lemma 13 for widespread requests. Using the notion of weak flexibility, we derive the following result. For an example of a configuration that is possible in this setting, but not in that of general flexibility, see (RC3) in Section 7.
As planar graphs without C 4 are not 3-degenerate, the following result further extends our attempts to attack Question 2. In this case K − 4 is allowed and we instead forbid the "house" C + 5 (a C 3 and a C 4 sharing an edge) and K 2,3 . Theorem 7. There exists ε > 0 such that every planar graph with neither C + 5 nor K 2,3 is weakly ε-flexible for lists of size 5. Planar graphs  Table 1: Known results in terms of degeneracy, flexibility, and choosability, including known lower bounds for choosability. Each graph class is a subclass of planar graphs. The first row indicates the forbidden subgraphs for each column. Results without a reference follow from some other entry in the table, including the degeneracy arguments implied by Theorem 1. Entries in blue are prior results that are not known to be tight, and entries in green are from this paper but are not known to be tight.
In Section 2, we develop the notation and prove Lemma 13, our main tool. We lay out our proof strategy in Section 3 and the proofs of Theorems 4, 5, 6, and 7 are in Sections 4, 5, 6, and 7, respectively.

Main tool
In this section, we strengthen and modify the main tool used in [13,14], in order to better understand the structural properties of a hypothetical minimum counterexample. Let 1 I denote the characteristic function of I, meaning that 1 I (v) = 1 if v ∈ I and 1 I (v) = 0 otherwise. For integer functions on the set of vertices of H, we define addition and subtraction in the natural way, adding and subtracting, respectively, their values at each vertex. Given a function f : . Given a set of graphs F and a graph H, a set I ⊂ V(H) is F -forbidding if the graph H together with one additional vertex adjacent to all of the vertices in I does not contain any graph from F . Let H be an induced subgraph of G and v ∈ V(H). We define deg H (v) as the degree of v measured in the induced subgraph H. 3 Note that the dodecahedron is a planar graph without C 3 , C 4 that is not 2-degenerate. The icosidodecahedron is a planar graph without C 4 that is not 3-degenerate. The truncated cube is a planar graph without C 4 , C 5 , C 6 , C 7 that is not 2-degenerate. The icosahedron is a planar graph that is not 4-degenerate. 4 We have not found a reference for this result and therefore include a proof for completeness as Observation 20 in Section 5 Definition 8 ((F , k)-boundary-reducibility). A graph H is an (F , k)-boundary-reducible induced subgraph of G if there exists a set B V(H) such that We define an additional reducibility condition wherein we weaken the (FIX) property in order to consider a new, weaker variation of flexibility.
In general, we may allow Fix(H) to be only one vertex, which clearly establishes this notion as a weaker one than that given in Definition 2. In both of the preceding definitions, we will sometimes refer to the set B as the boundary of the configuration.
Definition 10 ((F , k, b)-resolution). Let G be a graph with lists of size k that does not contain any graph in F as an induced subgraph. We define (F , k, b)-resolution of G as a set G i of nested subgraphs for 0 ≤ i ≤ M, such that G 0 : = G and where each H i is an induced (F , k)-boundary-reducible subgraph of G i−1 with boundary B i such that |V(H i ) \ B i | ≤ b and G M is an (F , k)-boundary-reducible graph with empty boundary and size at most b. For technical reasons, let G M+1 : = ∅.
A weak (F , k, b)-resolution is defined anaologously to an (F , k, b)-resolution; it uses weak (F , k)boundary-reducibility in the place of (F , k)-boundary-reducibility.
It is our goal to show that every graph that does not contain any graph from F as a subgraph contains a (weakly) reducible subgraph. Conceptually, we then think of a (weak) resolution as an inductively-defined object obtained by iteratively identifying some (weakly) reducible subgraph H with boundary B and deleting Going forward, when considering weak reducibility, let Fix(G) denote the union of each Fix(H) over the weak (F , k)-boundary-reducible subgraphs H in some resolution of G. While Fix(G) depends on the particular resolution under consideration, we will generally omit mention of the resolution when the context is clear. By definition, the following property holds: To prove weighted ε-flexibility, we use the following observation made by Dvořák et al. [14]. Lemma 11 ([14]). Let G be a graph and let L be a list assignment on V(G). Suppose G is L-colorable and there exists a probability distribution on L-colorings ϕ of G such that for every v ∈ V(G) and c ∈ L(v), Then G with L is weighted ε-flexible.
In light of lemma 11, we can derive a similar lemma for weak ε-flexibility, provided that the assumptions hold for sufficiently large and evenly distributed set of vertices.
Lemma 12. Let b be an integer. Let G be a graph with a widespread request and let L be a list assignment on V(G). Suppose G is L-colorable with a weak (F , k, b)-resolution and there exists a probability distribution on L-colorings ϕ of G such that for every v ∈ Fix(G) and The proof is very similar to proof of Lemma 11, but it makes use of the fact that requests are made for all vertices of G.
Proof. Let r be a widespread request for G and L. Let φ be chosen at random based on the given probability distribution. By assumption |Fix(G)| ≥ |V| b . By linearity of expectation: and thus there exists an L-coloring φ with ε · |V| b satisfied requests. Now, we are ready to strengthen the key lemma implicitly presented by Dvořák, Norin, and Postle in [14], and explicitly formulated as Lemma 4 in [13]).
Lemma 13. For all integers k ≥ 3 and b ≥ 1 and for all sets F of forbidden subgraphs there exists an ε > 0 as follows. Let G be a graph with an (F , k, b)-resolution. Then G with any assignment of lists of size k is weighted ε-flexible. Furthermore, if the request is widespread and G has weak (F , k, b)-resolution, then G with any assignment of lists of size k is weakly ε · 1 b -flexible.
Even though the statement of the lemma is noticeably stronger, its proof remains quite similar to the original formulation. We include the proof for the sake of completeness.
Proof. Let p = k −b and ε = p k−1 . For a graph G satisfying the assumptions and an assignment L of lists of size k, we prove the following claim by induction on the (F , k, b)-resolution: There exists a probability distribution on L-colorings ϕ of G such that (i) for every v ∈ Fix(G) and a color c ∈ L(v), the probability that ϕ(v) = c is at least ε, and (ii) for every color c and every F -forbidding set I in G of size at most k − 2, the probability that ϕ(v) c for all v ∈ I is at least p |I| .
Part (i) with Fix(G) = V(G) implies that G with L is weighted ε-flexible by Lemma 11. Part (i) with assumed widespread request implies that G with L is weakly ε · 1 b -flexible by Lemma 12.
The claim clearly holds for a graph with no vertices, the base case of the induction. Hence, suppose V(G i ) ∅. By the assumptions, there exists a subgraph H of G such that H is (F , k)-boundary-reducible. By definition, there exists a boundary set B ⊂ V(H), and let Q : = H − B. Moreover, by assumption, we know that the order of Q is at most b. By the induction hypothesis, there exists a probability distribution on L-colorings of G i+1 satisfying (i) and (ii). Choose an L-coloring ψ from this distribution and let L be the list assignment on has an L -coloring by (FORB) applied with I = ∅. Among all L -colorings of G[V(Q)], choose one uniformly at random, extending ψ to an L-coloring ϕ of G.
Let us first argue (ii) holds. Let I 1 = I \ V(Q) and I 2 = I ∩ V(Q). By the induction hypothesis, we have ϕ(v) c for all v ∈ I 1 with probability at least p |I 1 | . If I 2 = ∅, then this implies (ii). Hence, suppose has at most k b L -colorings, we conclude that the probability that ϕ(y) c for all y ∈ I 2 is at least 1/k b = p ≥ p |I 2 | . Hence, the probability that ϕ(y) c for all y ∈ I is at least p |I 1 |+|I 2 | ≥ p |I| , implying (ii).
Next, let us argue (i) holds. For v ∈ V(G) \ V(Q), this is true by the induction hypothesis. Hence, suppose that v ∈ V(Q), and let I be the set of neighbors of v in V(G) \ V(Q). Since G does not contain any graph from F but does contain the boundary B, and all vertices in I have a common neighbor, the set I is F -forbidden Hence, by the induction hypothesis, we have ψ(u) c for all u ∈ I with probability at least p k−2 . Assuming this is the case, (FIX) implies there exists an L -coloring of G[V(Q)] that gives v the color c. Since G[V(Q)] has at most k b L -colorings, we conclude that the probability that ϕ(v) = c is at least p k−2 /k b = ε. Hence, (i) holds.

Common proof preliminaries
We gather common definitions and reducible configurations for the forthcoming proofs. We also give an overview of the discharging method. Going forward, we will use the number of colors "available" to a vertex v in a configuration H as the maximum number of colors remaining in L(v) after coloring vertices exterior to the configuration (more precisely outside H − B). When considering (FIX) we reduce the number of available colors on a "fixed" vertex to 1, and when considering (FORB), we reduce the number of available colors on the vertices of a candidate set I by 1. (FIX): Each of u, v has two available colors. Fix a color ϕ(u) and choose an available color for ϕ(v) that is not ϕ(u) to extend the coloring. Fixing a color for v is symmetric.
If |I| ≥ 2, then I is not {K − 4 }-forbidding since connecting a new vertex to both u and v always creates a K − 4 . It remains to consider the cases where |I| = 1 but that is implied by (FIX).
We use the discharging method for the proofs of our theorems. We end this section with a brief overview of the method, for more detailed introduction to discharging method for graph coloring see [11]. Given a theorem we aim to prove, let H be a counterexample with the minimum number of vertices. Fix a plane embedding of H and let F(H) denote the set of faces of H. To each z ∈ V(H) ∪ F(H), assign an initial charge ch(z) so that the total sum is negative. In our proofs, this part is a standard and straightforward aplication of Euler's formula. We then redistribute the charge according to some discharging rules, which will preserve the total charge sum. Let ch * (z) denote the final charge at each z ∈ V(H) ∪ F(H). We recount the charge at this point and show that the final charge is non-negative for each vertex and face to conclude that the sum of the final charge is non-negative. This is a contradiction since the initial charge sum is negative and the discharging rules preserve the total charge sum. We conclude that a counterexample could not have existed.

Proof of Theorem 4
In this section we prove Theorem 4. Let The following is the only discharging rule: (D1) Every 5 + -vertex sends 1/2 to each incident 3-face.
We now check that each vertex and each face has non-negative final charge. Considering the vertices first, by (RC1) there is no 3 − -vertex. If v is a 4-vertex, then ch . Note that a vertex v is incident with at most deg(v) 2 3-faces since there is no K − 4 . Hence, each vertex has non-negative final charge.
We next turn our attention to faces, again with the goal of showing that their final charge is non-negative. If f is a 4 + -face, then ch * (v) = ch(v) = | f | − 4 ≥ 0, since f is not involved in the discharging rules. If f is a 3-face, then by (RC1) and (RC2), f is incident with at least two 5 + -vertices. By (D1), ch * ( f ) ≥ −1 + 2 · 1 2 = 0, completing the proof of Theorem 4.

Proof of Theorem 5
In this section, we prove Theorem 5. Let F consist of C 4 and all possible configurations such that C 3 's are in distance at most 1 and let G be a counterexample to Theorem 5 with the minimum number of vertices. Fix a plane embedding of G and note that by minimality, G must be connected. Let L be a list assignment on V(G) where each vertex receives at least four colors. The following configurations cannot appear in G: The discharging rules are the following: (D1) Every 3-vertex sends charge 1/3 to each incident face.
We now check that each vertex and each face has non-negative final charge.
Lemma 18. Each vertex has non-negative final charge.
Since to f by (D1) and (D2), respectively. Therefore, each vertex on f guarantees at least 2 3 to be sent to f , so ch * ( f ) ≥ −2 + 3 · 2 3 = 0. Suppose f is a 5-face, and let v 1 , v 2 , v 3 , v 4 , v 5 be the vertices on f in clockwise ordering. By (RC2), f cannot be incident with four 3-vertices. If f is incident with at most one 3-vertex, then f is incident with at least four 4 + -vertices. Since each 4 + -vertex on f sends at least 5 12 to f by (D2), ch * ( f ) ≥ −2 + 4 · 5 12 + 1 3 = 0. Therefore, we may assume f is incident with either two or three 3-vertices.
For completeness we include a proof of the following Observation 20 used in Table 1.
Observation 20. Every planar graph without C 4 such that the distance between C 3 's is at least 2 is 3degenerate.
Proof. Suppose to the contrary that there exists a planar graph with minimum degree at least 4 but neither C 4 nor C 3 distance at most 1. We use a simple discharging argument without any reducible configurations. Let the initial charge of each vertex v and each face f be deg(v) − 4 and | f | − 4, respectively. Therefore only 3-faces have negative initial charge. The discharging rules are the following (They will be applied in the order they are presented): • each 5 + -face sends 1/5 to each incident vertex.
• each vertex sends all its charge to its incident 3-face if it exists.
The final charge of each vertex remains non-negative. Each 5 + -face f has non-negative final charge since (| f | − 4)/| f |) ≥ 1/5. Let f be a 3-face and let v be a vertex on f . Each face incident with v except f is a 5 + -face, since there are neither C 4 nor C 3 distance at most 1. Since the minimum degree is at least 4, v receives charge at least 3 5 , all of which is sent to f . Thus, the final charge of f is at least −1 + 9/5 > 0.

Proof of Theorem 6
In this section, we prove Theorem 6. Let F = {C 4 , C 5 , C 6 } and let G be a counterexample with the minimum number of vertices. Fix a plane embedding of G and note that by minimality, G must be connected. Let L be a list assignment on V(G) where each vertex receives at least four colors. The following configurations cannot appear in G: In that case, we assign v 1 and v 2 distinct colors, leaving the two vertices on each 3-cycle with one and two available colors. The coloring can be extended by coloring the vertices with one available color first, then the remaining vertex on each 3-cycle.
As was the case previously, the case where |I| = 1 is implied by (FIX).
For each vertex v and each face f , let ch(v) = −2 and ch( f ) = | f | − 2. By Euler's formula the sum of initial charge is negative: Note that there are no 2 − -vertices by (RC1), so there are no 4-faces and no 5-faces. We remark that even though C 6 is forbidden, there might still be 6-faces; these can appear only in the form of two embedded 3-cycles.
The discharging rules are as follows: (D1) Every 6 + -face uniformly distributes its initial charge to every incident vertex.
(D2A) If f is incident with a 3-vertex, then f sends 4/7 to each incident 3-vertex and uniformly distributes its remaining charge to each incident 4-vertex.

Proof of Theorem 7
In this section, we prove Theorem 7. Let F = {C + 5 , K 2,3 } and let G be a counterexample with the minimum number of vertices. Fix a plane embedding of G and note that by minimality, G must be connected. Let L be a list assignment on V(G) where each vertex receives at least five colors. The following configurations cannot appear in G: If |I| ≥ 2, then I is not F -forbidding, since connecting a new vertex to any pair of vertices in H creates either a C + 5 or a K 2,3 . It remains to consider the cases where |I| = 1 but that is implied by (FIX). In the first and second case, we are left with a path with lists of size 1, 2, 2, 2 and 1, 2, 3, 1, respectively, in this order. In both cases, the coloring can be extended.
For each vertex v and each face f , let ch(v) = deg(v) − 2 and ch( f ) = −2. By Euler's formula the sum of initial charge is negative: The discharging rules are as follows: (D1) Every 6 + -vertex sends 2/3 to each incident face.
We now check that each vertex and each face has non-negative final charge.
Lemma 29. Each vertex has non-negative final charge.
Lemma 30. Each face has non-negative final charge.
Proof. If f is a 3-face, then in all situations each vertex on f gives 2/3 to f , so ch * ( f ) = −2 + 3 · 2 3 = 0. If f is a 4-face, then f is incident with at most two 4-vertices by (RC2). We distinguish the following cases based on the number of 4-vertices incident with f .

Conclusions
One can see Lemma 13 and, in particular, definition of (weak) (F , k, b)-resolution as a generalization of degeneracy order. There, the role of single vertices is replaced by (weak) (F , k)-boundary-reducibile configurations. The resolution can be easily constructed in polynomial time under a mild assumption that the size of the boundary of each reducible configuration is bounded (which is the case of the theorems it is applied to in this paper). Using the resolution, the list coloring satisfying any request for a single vertex only can be obtained straightforwardly. However, it is not clear how to reconstruct ε-satisifable coloring for the given request (using the resolution or not) even though the existence of such coloring is guaranteed by Lemma 13. As the flexibility concept has a substantial algorithmic motivation, it would be very interesting to explore its algorithmic potential. Besides the open questions given in the introduction, we propose some open areas of inquiry. In addition to further exploring the notion of weak flexibility, we propose two possible directions that align with the general effort to distinguish between flexibility and choosability in the class of planar graphs.
First, Cohen-Addad, Hebdige, Král', Li, and Salgado [10] constructed a planar graph with neither C 4 nor C 5 that is not even 3-colorable, refuting Steinberg's Conjecture (see [25]). In this vein, we feel it would be interesting to determine whether it is possible to strengthen Theorem 6 to graphs without C 4 and C 5 with lists of size 4.
As pointed out in [14] it would be nice to narrow the gap between d-degenerate graphs and (weighted) ε-flexible graphs with lists of size k. Theorem 1 shows that d-degenerate graphs are ε-flexible when the size of the lists is at least d + 2, but the bound conjectured in [14] is d + 1. We propose a study of outer-planar graphs, which are 2-degenerate, but it is not known whether lists of size 3 suffice to achieve ε-flexibility.