In a proper graph coloring, we want to assign to each vertex of a graph one of a fixed number of colors in such a way that adjacent vertices receive distinct colors. Dvořák, Norin, and Postle  (motivated by a similar notion considered by Dvořák and Sereni ) introduced the following graph coloring question called Flexibility. If some vertices of the graph have a preferred color, is it possible to properly color the graph so that at least a constant fraction of the preferences are satisfied? As it turns out, this question is trivial in the ordinary proper coloring setting with a bounded number of colors (-coloring). The answer is always positive since we can permute the colors according to the request and therefore satisfy at least fraction . On the other hand, Flexibility brought about a number of interesting problems in the list coloring setting.
A list assignment for a graph is a function that to each vertex assigns a set of colors, and an -coloring is a proper coloring such that for all . A graph is -choosable if is -colorable from every assignment of lists of size at least . A weighted request is a function that to each pair with and assigns a nonnegative real number. Let . For , we say that is -satisfiable if there exists an -coloring of such that
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 with a list assignment is a function with such that for all . For , a request is -satisfiable if there exists an -coloring of such that for at least vertices .
Note that in particular, a request is -satisfiable if and only if the precoloring given by extends to an -coloring of . We say that a graph with the list assignment is -flexible if every request is -satisfiable, and it is weighted -flexible if every weighted request is -satisfiable.
Dvořák, Norin, and Postle  established the basic properties of the concept. They prove several theorems in terms of degeneracy and maximum average degree. For example: For every , there exists such that -degenerate graphs with assignment of lists of size are weighted -flexible. Those results imply structural theorems for planar graphs:
There exists such that every planar graph with an assignment of lists of size is -flexible.
There exists such that every planar graph of girth at least five with an assignment of lists of size is -flexible.
There exists such that every planar graph of girth at least 12 with an assignment of lists of size is weighted -flexible.
Those results prompted a number of interesting questions. The main meta-question for planar graphs is whether such bounds can be improved to match the choosability. Notice that choosability is a lower bound for the minimum size of lists in the statement. Dvořák, Masařík, Musílek, and Pangrác subsequently answer two such questions. In  they show that triangle-free planar graphs with an assignment of lists of size are weighted -flexible. This is optimal since there are triangle-free planar graphs that are not 3-choosable [5, 11]. In  they show that planar graphs of girth at least six with an assignment of lists of size are weighted -flexible. There is still a small gap left open since even planar graphs of girth at least 5 are 3-choosable. The biggest question in this direction that is still unanswered is stated as follows.
Does there exist such that every planar graph and assignment of lists of size five is (weighted) -flexible?
This would be optimal in terms of choosability [10, 9]. However, (if it is true) it might be difficult to obtain such a result since even the result of Thomassen  for choosability is very involved. In particular, compare it to a rather easy proof  for choosability of triangle-free planar graphs and still the respective result for flexibility  was quite technical.
In this paper, we propose a step towards answering Question 1 by proving the following theorem.
There exists such that each planar graph without 4-cycles with an assignment of lists of size five is weighted -flexible.
Since planar graphs without 4-cycles are 4-choosable  there is a gap left open.
We say that a face is edge-adjacent to another face if both share an edge. Since graphs we are dealing with does not contain 4-cycles they cannot contain two edge-adjacent triangles.
Let be a graph. For a positive integer , a set is -independent if the distance between any distinct vertices of in is greater than . Let
denote the characteristic function of, i.e., if and otherwise. For functions that assign integers to vertices of , we define addition and subtraction in a natural way, adding/subtracting their values at each vertex independently. For a function and a vertex , let denote the function such that for and . A list assignment is an -assignment if for all .
Suppose is an induced subgraph of another graph . For an integer , let be defined by for each . For another integer , we say that is a -reducible induced subgraph of if
for every , is -colorable for every -assignment , and
for every -independent set in of size at most , is -colorable for every -assignment .
Note that (FORB) in particular implies that for all . Intuitively, (FIX) requires that is -colorable even if we prescribe the color of any single vertex of , and (FORB) requires that is -colorable even if we forbid to use one of the colors on the set .
For all integers , there exists as follows. If for every , the graph contains an induced -reducible subgraph with at most vertices, then with any assignment of lists of size is weighted -flexible.
3 Reducible configurations
In view of Lemma 3, we aim to prove that every planar graph without 4-cycles contains a -reducible induced subgraph with the bounded number of vertices.
In any graph , a vertex of degree at most forms a -reducible subgraph.
From now on suppose that the minimum degree of is . We describe one more easy reducible configuration (see Figure 1) that, in combination with discharging, turns out to be sufficient to derive the promised theorem.
If is a planar graph without 4-cycles, then a vertex together with neighbors of degree four forms a -reducible configuration on vertices.
(FIX): If vertex has fixed color then we have enough remaining colors on its neighbors to complete the coloring. If any other vertex is fixed then it crosses out one color from and in case has a neighbor in it also crosses out one of its colors. In both cases, we can set a color of and complete the coloring greedily.
(FORB): Observe that if we forbid a color of a vertex that is not adjacent to any vertex in then its color is determined and therefore it crosses out one color of . The same effect has a forbidden color of . If we forbid a color of a vertex such that it forms a triangle then it does not force anything unless vertex has also a forbidden color. In the latter case two colors are crossed out from the list of . Keep in mind that this cannot happen twice since there are no two edge-adjacent triangles. Since only three colors are removed from the list of , we can color and then the rest of the graph greedily to conclude the proof. ∎
Let us assign charge to each vertex and charge to each face of , where denotes the length of the facial walk of . By Euler’s formula, we have .
Note that only triangle-faces have a negative charge before any redistribution of the charge. We redistribute the initial charge according to the following rules.
For each face of if then sends to any edge-adjacent triangle-face.
For each vertex of if then sends to each adjacent triangle-face.
For each vertex of and each its incident face if and then sends to .
Observe that the charge of any face does not drop below zero by Rule (R1). Any vertex of degree at least 5 with incident triangle-faces sends at most by Rule (R2) and by Rule (R3). View that the number of incident triangle-faces for a single vertex is at most because there are not any edge-adjacent triangle-faces. Therefore . It follows that .
It remains to argue that all triangle-faces obtain enough of the charge. Each of them receives the charge at least by Rule (R1). If one of its vertices has degree at least five we are done by Rule (R2). Therefore all of them have degree exactly four. We call such triangle-face poor.
We do one more redistribution of charge.
For each poor triangle-face of and for each edge-adjacent face if then sends to .
A Combination of Rules (1) and (4) yields that poor triangle-faces have a positive charge. They obtain by Rule (1) and three times by Rule (4). Finally, we show that the charge of larger faces remains non-negative after the application of Rule (R4).
Consider face of length at least five edge-adjacent to some triangle-face . Recall that . By Lemma 5 applied on vertex we claim that . By the same argument repeated on vertex we derive . Therefore, by Rule (R3) receive charge at least from each and . This, combined with an observation that has at most edge-adjacent poor triangle-faces, yields the promised claim for . Larger faces sent only by Rule (R1) altogether and therefore they can pay an additional .
This is a contradiction with the original negative assignment of charge and therefore we derive Theorem 2.
We proved that planar graphs without 4-cycles are weighted -flexible for lists of size at least five. This is a middle step to answer Question 1 that might be challenging as mentioned in the introduction. Based on the proof possible difficulties we suggest, as a next step to prove the conjecture, to inspect first planar graphs without diamonds ().
There exists such that every planar graph without diamonds and assignment of lists of size five is (weighted) -flexible.
Another possible direction is closing the gap between flexibility and choosability for planar graphs without 4-cycles.
Does there exists such that every planar graph without 4-cycles and assignment of lists of size four is (weighted) -flexible?
I would like to thank Zdeněk Dvořák for helpful comments.
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