Flexibility of planar graphs of girth at least six

02/11/2019 ∙ by Zdeněk Dvořák, et al. ∙ 0

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.

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1 Introduction

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 [2] (motivated by a similar notion considered by Dvořák and Sereni [3]) introduced the following graph coloring question: 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 is satisfied? As it turns out, this question is trivial in the ordinary proper coloring setting with a bounded number of colors (the answer is always positive [2]), but in the list coloring setting this gives rise to a number of interesting problems.

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 (say for at most one color , and for any other color ): 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 (of course, weighted -flexibility implies -flexibility).

Dvořák, Norin and Postle [2] established basic properties of the concept and proved that several interesting graph classes are flexible:

  • For every , there exists such that -degenerate graphs with assignments of lists of size are weighted -flexible.

  • There exists such that every planar graph with assignment of lists of size is -flexible.

  • There exists such that every planar graph of girth at least five with assignment of lists of size is -flexible.

They also raised a number of interesting questions, including the following one.

Problem 1.

Does there exists such that every planar graph and assignment of lists of size

  • five in general,

  • four if is triangle-free,

  • three if has girth at least five

is (weighted) -flexible?

Let us remark that the proposed list sizes match the best possible bounds guaranteeing the existence of a coloring from the lists [6, 7]. In [1], we gave a positive answer to the part (b).

In this note, we consider coloring from lists of size three, cf. the part (c). We prove that girth at least six is sufficient to ensure flexibility with lists of size three; this improves upon the result of Dvořák, Norin and Postle [2] who proved that girth at least is sufficient.

Theorem 2.

There exists such that each planar graph of girth at least six with assignment of lists of size three is weighted -flexible.

Note that while it is trivial to prove that planar graphs of girth at least six are 3-choosable, Theorem 2 is substantially more complicated to establish. As the proof that planar graphs of girth five are 3-choosable [4] is quite involved, we suspect proving the part (c) of Problem 1 (if true) would be hard.

2 Preliminaries

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 the 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 . Before we proceed, let us give an intuition behind these definitions. Consider any assignment of lists of size to vertices of . The function describes how many more (or fewer) available colors each vertex has compared to its degree. Suppose we -color , and let be the list assignment for obtained from by removing from the list of each vertex the colors of its neighbors in . In , each vertex has at least available colors, since each color in corresponds to a neighbor of in . Hence, (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 -independent set .

The following lemma is implicit in Dvořák et al. [2] and appears explicitly in [1].

Lemma 3.

For all integers and , there exists as follows. Let be a graph of girth at least . 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.

We also use the following well-known fact.

Lemma 4 (Thomassen [5]).

Let be a connected graph and a list assignment such that for all . If either there exists a vertex such that , or some -connected component of

is neither complete nor an odd cycle, then

is -colorable.

3 Reducible configurations

In view of Lemma 3, we aim to prove that every planar graph of girth at least contains a -reducible induced subgraph (with bounded number of vertices). Note that in the case of -reducibility, the -independent set considered in (FORB) has size at most . Hence, (FORB) is implied by (FIX) and the additional assumption that , i.e., that , for every . In particular, the value of is not important, and for brevity we will say -reducible instead of -reducible.

Observation 5.

In any graph , a vertex of degree at most forms a -reducible subgraph.

Let be a subgraph of a graph . A -nice path in is a path whose internal vertices have degree exactly three in and do not belong to . A vertex is -nicely reachable from a vertex if there exists a -nice path from to ; let denote the length of shortest such path. If is not -nicely reachable from , let us define .

Lemma 6.

Let be a graph, let be a subgraph of , and let be distinct vertices of degree two. If is -nicely reachable from , then contains a -reducible induced subgraph disjoint from with at most vertices.

Proof.

Let be a shortest -nice path from to in . Clearly, and is an induced path in disjoint from . We claim that is -reducible. Note that for every , and thus it suffices to show that satisfies (FIX). For any vertex of , consider a -assignment . Each vertex of other than has list of size at least two, and thus we can -color the path greedily starting from . Hence, is -colorable; and thus (FIX) holds. ∎

Lemma 7.

Let be a graph, let be a subgraph of , let be a vertex of degree , and let be distinct vertices of degree two. If all vertices , …, are -nicely reachable from , then contains a -reducible induced subgraph disjoint from with at most vertices.

Proof.

For , let be a shortest -nice path from to in . Let . We can assume that every two of the paths intersect only in and that is an induced subgraph of , as otherwise the claim follows by Lemma 6. Note that and .

We claim that is -reducible. Note that for every , and thus it suffices to show that satisfies (FIX). For any vertex of , consider a -assignment . Each vertex of other than has list of size at least two, and thus we can -color the tree greedily starting from . Hence, is -colorable; and thus (FIX) holds. ∎

Lemma 8.

Let be a -connected plane graph of girth at least six, and let and be distinct -faces of sharing at least one edge such that all vertices incident with or have degree at most three. If is incident with a vertex of degree two, then contains a -reducible induced subgraph with at most vertices, all incident with or .

Proof.

Let and be the cycles bounding and in , respectively. Since has girth at least six, is a path. Let be the subgraph of induced by vertices incident with or ; we have .

We claim that is -reducible. Note that for every , and thus it suffices to show that satisfies (FIX). For any vertex of , consider a -assignment . Let be a color in and let be the list assignment for obtained from by removing from the lists of neighbors of . Note for all , and if . Observe that is connected, and that if , then contains an even cycle. Hence is -colorable by Lemma 4, and thus is -colorable. We conclude that (FIX) holds. ∎

Lemma 9.

Let be a -connected plane graph of girth at least six, and let , , and be distinct -faces of , all incident with a common vertex. If all vertices incident with these faces have degree at most three, then contains a -reducible induced subgraph with at most vertices, all incident with , , or .

Proof.

Let be the subgraph of induced by vertices incident with , , or ; we have . Since has girth at least six, the boundaries of any two of the faces intersect in a path. By Lemma 8, we can assume that all vertices of have degree exactly three in ; consequently, any two of the faces , , and share exactly one edge.

We claim that is -reducible. Note that for every , and thus it suffices to show that satisfies (FIX). For any vertex of , consider a -assignment . Let be a color in and let be the list assignment for obtained from by removing from the lists of neighbors of . Note for all . Observe that is connected and contains an even cycle. Hence is -colorable by Lemma 4, and thus is -colorable. We conclude that (FIX) holds. ∎

4 Discharging

Let be a -connected plane graph of girth at least six such that and every -cycle bounds a face, and let be either a vertex or a -cycle in . Let us assign the initial charge to each vertex and to each face of . If consists of a single vertex , then let , otherwise let for each . Let if and if . By Euler’s formula, we have

For a face of and a vertex , let us define as if is incident with , and as the minimum of over all vertices of degree three incident with otherwise. Let us now redistribute the charge according to the following rules:

  • Each vertex sends 2 units of charge to each vertex of degree two such that .

  • Each vertex of degree at least sends 1 unit of charge to each vertex of degree two such that .

  • Each face of length at least sends 1 unit of charge to each vertex of degree two such that receives at most one unit of charge by (T0) and (T1) and .

Let denote the final charge after performing the redistribution according to these rules.

Whenever one of the rules (T0), (T1), or (T2) applies, this is because of some shortest -nice path ending in a vertex of degree two and starting in a vertex of , or a vertex of degree at least four, or a vertex incident with a -face. This path is not necessarily unique, but for each application of a rule we fix one such path arbitrarily; and if has length at least one and starts with an edge , we say that the charge to is being sent through the edge .

Lemma 10.

Let be a -connected plane graph of girth at least six such that and every -cycle bounds a face, and let be either a vertex or a -cycle in . If does not contain a -reducible induced subgraph with at most vertices disjoint from , then for all and for all . If additionally is a -cycle, then some vertex or face has positive final charge.

Proof.

Let be a face of . Since is -connected, , and has girth at least six, we have . If , then . Hence, suppose that . Let denote the cycle bounding , and consider a vertex , contained in a subpath of . If and do not belong to and have degree three, then let ; if belongs to or has degree other than three, then ; otherwise, belongs to or has degree other than three, and we let . For each vertex of degree two to which sends charge according to (T2), let if is incident with and let be a vertex of degree three incident with such that otherwise. Observe that all internal vertices of other than have degree three and no vertex of belongs to , as otherwise either contains a -reducible induced subgraph with at most vertices disjoint from by Lemma 6, or receives units of charge by (T0) and (T1). Furthermore, if sends charge to distinct vertices and by (T2), then the paths and are edge-disjoint, as otherwise would contain a -reducible induced subgraph with at most vertices disjoint from by Lemma 6. We conclude that the amount of charge sent by is at most , and thus .

Furthermore, if shares an edge with , then the amount of charge sent by is at most by the same argument, and thus . If , this implies . If , then observe that does not send charge by (T2): if it sent charge to some vertex , then the endvertices of would have distinct neighbors in , and thus would receive at least units of charge by (T0) and (T1), which is a contradiction. We conclude that if a face of length at least shares an edge with , then .

Let be a vertex of . Since is -connected and , we have . Let us first consider the case that or . Let denote the number of vertices of degree two such that . Note that does not send charge to two distinct vertices and through the same edge, as otherwise we would have and by Lemma 6 would contain a -reducible induced subgraph disjoint from with at most vertices. Hence, we have . Furthermore, no charge is sent through the edges of , and thus if and is a -cycle, then . Finally, if and , then Lemma 7 implies , as otherwise would contain a -reducible induced subgraph disjoint from with at most vertices. Hence, if consists just of the vertex , then by (T0); if is a -cycle and , then by (T0); if and , then by (T1); and if and , then by (T1).

Furthermore, consider the case that shares an edge with a -face , bounded by a -cycle . Since has girth at least , is a path; let and be its endvertices, and let and be the edges of incident with and , respectively. If charge is sent through and to vertices and , respectively, then either , or and the path contains a vertex of degree at least by Lemma 6. In the former case, . In the latter case we can by symmetry assume that is at distance two from , and thus also receives charge from by (T1) and .

We conclude that if shares an edge with a -face , then contains a vertex with (with if charge is sent through both and , and otherwise). Together with the preceding analysis of the case that a -face shares an edge with , this implies that if is a -cycle, then some vertex or face has positive final charge.

Finally, let us consider the final charge of a vertex of degree at most three. If , then ; hence, suppose . If receives charge by (T0), then . Hence, we can assume this is not the case, and thus for every . Let and be the faces incident with . If receives charge neither from and by (T2) nor from the vertices incident with or by (T1), then both and would have length six, and either would contain a -reducible induced subgraph with at most vertices disjoint from by Lemma 6, or all all other vertices incident with and would have degree exactly three and would not belong to . But in the latter case, would contain a -reducible induced subgraph with at most vertices disjoint from by Lemma 8. This is a contradiction, and thus receives at least one unit of charge from , , or their incident vertices.

If receives charge from at least two vertices or faces, then . Hence, we can assume that receives charge either from exactly one vertex or from exactly one face. In particular, we can by symmetry assume that ; let be the cycle bounding . By Lemma 6, the neighbors of have degree at least three. Suppose now that does not receive charge from either of its neighbors. We can by symmetry assume that receives charge from or a vertex incident with . In the latter case, we have , and since has girth at least six, we conclude that is not incident with . Consequently, Lemma 6 implies that all vertices incident with other than have degree exactly three. Let be the neighbor of distinct from and . By symmetry, we can assume that , and thus has degree exactly three. Let be the face incident with the path . Since does not receive charge from , we have . Since has girth at least six, is not incident with , and since does not receive charge from a vertex other than , Lemma 6 implies that all vertices incident with have degree exactly three. However, then would contain a -reducible induced subgraph with at most vertices disjoint from by Lemma 8.

Therefore, we can assume that receives charge from its neighbor, by symmetry say , and thus , and by Lemma 6, all vertices incident with or other than and have degree three. Let be the neighbor of distinct from and , let denote the face incident with the path , and let and be the faces incident with and , respectively, distinct from , , and . Since does not receive charge from , , or , we conclude these faces have length six, and since has girth at least six, is not incident with , , or . Consequently, all vertices incident with these faces have degree three. Let and be the edges shares with and , and let and be the faces incident with and , respectively, distinct from , , and . Since does not receive charge from and , both faces have length six. Let be the edge shared by and . Since has girth at least six and every -cycle in bounds a face, we conclude , and consequently is incident with at most one of the faces and . By symmetry, assume that is not incident with . By Lemma 6 and the assumption that receives charge only from , we conclude that all vertices incident with have degree three. However, then contains a a -reducible induced subgraph with at most vertices disjoint from (induced by the vertices incident with , , and ) by Lemma 9. This is a contradiction, showing that . ∎

We are now ready to prove the main result of this note.

Proof of Theorem 2.

Let be a plane graph of girth at least six. We apply Lemma 3 (with , and ) to show that is weighted -flexible (for fixed corresponding to the given values of , , and ) with any assignment of lists of size . Since every subgraph of is planar and has girth at least six, it suffices to prove that contains a -reducible induced subgraph with at most vertices. Without loss of generality, we can assume that is connected, and by Observation 5, we can assume that all vertices of have degree at least two.

Let be a -connected block of ; since has minimum degree at least two, we have . If , then let be the subgraph of consisting of the vertex in that intersects the rest of the graph ; otherwise, let be an arbitrary subgraph of consisting of a single vertex. Without loss of generality, we can assume that the vertex of is incident with the outer face of .

If all -cycles in bound faces, then let and . Otherwise, let be a -cycle in such that the open disk bounded by is not a face of , and is minimal among the cycles with this property. Let be the subgraph of drawn in the closure of . Note that is -connected and every -cycle in bounds a face.

Let be the assignment of charges to vertices and faces of described at the beginning of the section. Recall that the sum of these charges is negative if and if is a -cycle. Redistributing the charge according to the rules (T0)—(T2) gives us the charge assignment with the same sum of charges, and thus either some vertex or face has negative final charge, or is a -cycle and all vertices and faces have final charge . Lemma 10 implies that contains a -reducible induced subgraph disjoint from with at most vertices. Since is disjoint from , is also a -reducible induced subgraph of . ∎

References