Packing and covering odd cycles in cubic plane graphs with small faces
Abstract.
We show that any connected cubic plane graph on vertices, with all faces of size at most , can be made bipartite by deleting no more than edges, where and are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria, Klein and Stehlík [SIAM J. Discrete Math. 26 (2012) 1458–1469].
1. Introduction
A set of edges intersecting every odd cycle in a graph is known as an odd cycle (edge) transversal, or odd cycle cover, and the minimum size of such a set is denoted by . A set of edgedisjoint odd cycles in a graph is called a packing of odd cycles, and the maximum size of such a family is denoted by . Clearly, . Dejter and NeumannLara [6] and independently Reed [17] showed that in general, cannot be bounded by any function of , i.e., they do not satisfy the Erdős–Pósa property. However, for planar graphs, Král’ and Voss [14] proved the (tight) bound .
In this paper we focus on packing and covering of odd cycles in connected cubic plane graphs with all faces of size at most . Such graphs—and their dual triangulations—are a very natural class to consider, as they correspond to surfaces of genus of nonnegative curvature (see e.g. [21]).
A muchstudied subclass of cubic plane graphs with all faces of size at most is the class of fullerene graphs, which only have faces of size and . Faria, Klein and Stehlík [9] showed that any fullerene graph on vertices has an odd cycle transversal with no more than edges, and characterised the extremal graphs. Our main result is the following extension and sharpening of their result to all connected cubic plane graphs with all faces of size at most .
Theorem 1.1.
Let be a connected cubic plane graph on vertices with all faces of size at most , with pentagonal and triangular faces. Then
In particular, always holds, with equality if and only if all faces have size and , for some , and .
If is a fullerene graph, then and Euler’s formula implies that , so Theorem 1.1 does indeed generalise the result of Faria, Klein and Stehlík [9]. We also remark that the smallest connected cubic plane graph with all faces of size at most achieving the bound in Theorem 1.1 is the ubiquitous buckminsterfullerene graph (on 60 vertices).
The rest of the paper is organised as follows. In Section 2, we introduce the basic notation and terminology, as well as the key concepts from combinatorial optimisation and topology. In Section 3, we introduce the notions of patches and moats, and prove bounds on the area of moats. Then, in Section 4, we use these bounds to prove an upper bound on the maximum size of a packing of cuts in triangulations of the sphere with maximum degree at most . Using a theorem of Seymour [19], we deduce, in Section 5, an upper bound on the minimum size of a join in triangulations of the sphere with maximum degree at most , and then dualise to complete the proof of Theorem 1.1. In Section 6, we deduce lower bounds on the size of a maximum cut and a maximum independent set in connected cubic plane graphs with no faces of size more than . Finally, in Section 7, we show why the condition on the face size cannot be relaxed, and briefly discuss the special case when the graph contains no pentagonal faces.
2. Preliminaries
Most of our graphtheoretic terminology is standard and follows [1]. All graphs are finite and simple, i.e., have no loops and parallel edges. The degree of a vertex in a graph is denoted by . If all vertices in have degree , then is a cubic graph. The set of edges in with exactly one end vertex in is denoted by . A set of edges is a cut of if , for some . When there is no risk of ambiguity, we may omit the subscripts in the above notation.
The set of all automorphisms of a graph forms a group, known as the automorphism group . The full icosahedral group is the group of all symmetries (including reflections) of the regular icosahedron. The full tetrahedral group is the group of all symmetries (including reflections) of the regular tetrahedron.
A polygonal surface is a simply connected manifold, possibly with a boundary, which is obtained from a finite collection of disjoint simple polygons in by identifying them along edges of equal length. We denote by the union of all polygons in , and remark that is a surface.
Based on this construction, may be viewed as a graph embedded in the surface . Accordingly, we denote its set of vertices, edges, and faces by , , and , respectively. If every face of is incident to three edges, is a triangulated surface, or a triangulation of . In this case, can be viewed as a simplicial complex. If is a simplicial complex and , then is the subcomplex induced by , and is the subcomplex obtained by deleting and all incident simplices. If is a subcomplex of , then we simply write instead of .
If is a graph embedded in a surface without boundary, the dual graph is the graph with vertex set , such that if and only if and share an edge in . The size of a face is defined as the number of edges on its boundary walk, and is denoted by . Note that .
Any polygonal surface homeomorphic to a sphere corresponds to a plane graph via the stereographic projection. Therefore, terms such as ‘plane triangulation’ and ‘triangulation of the sphere’ can be used interchangeably. We shall make the convention to use the term ‘cubic plane graphs’ because it is so widespread, but refer to the dual graphs as ‘triangulations of the sphere’ because it reflects better our geometric viewpoint.
Given a polygonal surface , the boundary is the set of all edges in which are not incident to two triangles; the number of edges in the boundary is denoted by . With a slight abuse of notation, will also denote the set of vertices incident to edges in . The set of interior vertices is defined as .
Given a triangulated surface , we define to be the number of faces in , and the combinatorial curvature of as . Recall that the Euler characteristic of a polygonal surface is equal to . It can be shown that is a topological invariant: it only depends on the surface , not on the polygonal decomposition of . If is any contractible space, then , and if is the standard dimensional sphere, then . The following lemma is an easy consequence of Euler’s formula and double counting, and we leave its verification to the reader.
Lemma 2.1.
Let be a triangulated surface with (a possibly empty) boundary . Then
We remark that, if we multiply both sides of the equation by , we obtain a discrete version of the Gauss–Bonnet theorem (see e.g. [15]), where the curvature is concentrated at the vertices.
In order to prove Theorem 1.1, it is more convenient to work with the dual graphs, which are characterised by the following simple lemma. The proof is an easy exercise, which we leave to the reader.
Lemma 2.2.
If is a connected simple cubic plane graph with all faces of size at most , then the dual graph is a simple triangulation of the sphere with all vertices of degree at least and at most .
We will use the following important concept from combinatorial optimisation. Given a graph with a distinguished set of vertices of even cardinality, a join of is a subset such that is equal to the set of odddegree vertices in . The minimum size of a join of is denoted by . When is the set of odddegree vertices of , a join is known as a postman set. A cut is an edge cut such that is odd. A packing of cuts is a disjoint collection of cuts of ; the maximum size of a packing of cuts is denoted by .
3. Patches and moats
From now on assume that is a triangulation of the sphere with all vertices of degree at most . We define a subcomplex to be a patch if in the dual complex , the faces corresponding to form a subcomplex homeomorphic to a disc. (Equivalently, one could say that is a patch if is an induced, contractible subcomplex of .) A patch such that is called a patch. We remark that a patch has combinatorial curvature if and only if all vertices in the boundary have degree in . If has degree , and the set of vertices at distance at most from contains only vertices of degree , then the patch is denoted by . The subcomplex of the dual complex formed by the faces in is denoted by ; see Figure 3.1.
The following isoperimetric inequality follows from the work of Justus [13, Theorem 3.2.3 and Table 3.1]^{1}^{1}1Gunnar Brinkmann [2] has pointed out an error in the statement and proof of [8, Lemma 4.4] on which [13, Theorem 3.2.3] is based, but has sketched a different way to prove [13, Theorem 3.2.3]..
Lemma 3.1 (Justus [13]).
Let be a polygonal surface homeomorphic to a disc, with all internal vertices of degree and with faces, all of size at most . Let , and suppose that . Then
Equality holds if , for some integer , and only if at most one face in has size less than .
Proof.
The minimum possible values of are given in [13, Table 3.1], for all possible numbers of hexagonal, pentagonal, square, and triangular faces. In each case, our bound is satisfied. Moreover, it can be checked that equality holds only if at most one face in has size less than . Finally, if , then it can be shown that and . Hence, . ∎
We can use Lemma 3.1 to deduce the following isoperimetric inequality for triangulations. Certain special cases of the inequality were already proved by Justus [13].
Lemma 3.2.
Let be a triangulation of the sphere with all vertices of degree at most , and let be a patch of combinatorial curvature . Then
Equality holds if , for some integer , and only if at most one vertex in has degree less than .
Proof.
Put , and let be the subcomplex of formed by the faces corresponding to . By Lemma 3.1,
(3.1) 
Moreover, the following two equalities were shown by Justus [13, equations (3.8) and (3.11)]
(3.2) 
(3.3) 
So, combining (3.1), (3.2) and (3.3) gives
(3.4) 
Equality holds in (3.4) if and only if equality holds in (3.1). The latter is true only if at most one face in has size less than , or equivalently, only if at most one vertex in has degree less than . For the final part, it is enough to note that if , then , so equality holds in (3.1) and therefore in (3.4). ∎
Let be a patch. A moat of width in surrounding is the set of all the faces in with at least one vertex in . More generally, we can define a moat of width in surrounding recursively as . With a slight abuse of notation, will also denote the subcomplex of formed by the faces in . If is a patch, then is a moat of width surrounding . See Figure 3.2 for an example of a moat.
Under certain conditions, the area of a moat can be bounded in terms of , , and .
Lemma 3.3.
Let be a triangulation of the sphere with maximum degree at most , and suppose is a patch, for some . If is a patch, for every , then
Equality holds if , for some integer , and only if at most one vertex in has degree less than .
Proof.
The dual complex is homeomorphic to the sphere and the subcomplex formed by the faces corresponding to is homeomorphic to a disc, so by the Jordan–Schoenflies theorem, the subcomplex formed by the faces corresponding to is also homeomorphic to a disc. Hence, is also a patch. Moreover, has Euler characteristic , so by Lemma 2.1, . Therefore, , i.e., is a patch. Applying (3.5) to and to ,
Hence, , so by induction, and the fact that is a patch for all ,
(3.6) 
so the area of is
The combinatorial curvature of is at most , so by Lemma 3.2,
with equality if , for some integer , and only if at most one vertex in has degree less than . ∎
4. Packing odd cuts in triangulations of the sphere with maximum degree at most
We now relate certain special types of packings of cuts to packings of ,  and moats.
Lemma 4.1.
Let be a triangulation of the sphere with all vertices of degree at most , and let be the set of odddegree vertices in . There exists a family on and a vector with the following properties.

is a packing of moats in ;

The total width of is ;

For every , the subcomplex is a patch;

Every is a , , or moat in ;

If is an inclusionwise minimal element in , then ;

is laminar.
Proof.
Consider a packing of inclusionwise minimal cuts in of size of . Note that is odd, for every . Since and , we can assume that ; otherwise we could replace by in . Finally, we can also assume that, subject to the above conditions, minimises .
We remark that is a laminar family. Indeed, suppose that , , and . Then , so there is a face of in . Since
it follows that , contradicting the fact that is a packing of cuts. Hence, is laminar.
We summarise the properties of the family below.

is a packing of cuts;

;

is an inclusionwise minimal cut, for every ;

for all ;

is laminar.
We let be the subfamily of consisting of the elements such that
for every such that . For each , let
and let .
To prove 1, we use an argument very similar to the one we used to prove 6. Clearly, for every , is a moat around of width . Let , and suppose that . Then there exists a face and sets such that , , and
But then , so by 1, . Hence, by the construction of , . This proves 1.
Lemmas 3.3 and 4.1 can be used to prove the following upper bound on the maximum size of a packing of odd cuts in spherical triangulations with all vertices of degree at most , which may be of independent iterest. By taking the planar dual, we also get an upper bound on for the class of connected cubic plane graphs with all faces of size at most .
Theorem 4.2.
Let be a triangulation of the sphere with maximum degree at most . If is the set of odddegree vertices of , then
In particular, always holds, with equality if and only if all vertices have degree and , for some , and .
Proof.
Let be a packing of ,  and moats in of total width , as guaranteed by Lemma 4.1. Let be the total area of moats of , where . Define the incidence vectors as follows: for every , let , , be the width of the moat, moat and moat surrounding , respectively.
Define the inner product on by and the norm by . With this inner product, the total width of ,  and moats in can be expressed as , , and , respectively. Therefore,
(4.1) 
To prove the inequality in Theorem 4.2, we compute lower bounds on , and in terms of the vectors , and , and then use the fact that the moats are disjoint, so the sum cannot exceed , the number of faces of . Simplifying the inequality gives the desired bound.
To bound , recall that by property 5 of Lemma 4.1, every moat in is of the form , where is a vertex in . By Lemma 3.3,
(4.2) 
and summing over all moats gives the equality
(4.3) 
To bound , let be a nonempty moat in , for some . By the laminarity of , the patch contains the (possibly empty) moats , for all vertices . All the moats are pairwise disjoint, so by (4.2) and the Cauchy–Schwarz inequality,
Hence, by Lemma 3.3,
(4.4) 
Summing over all moats gives the inequality
(4.5) 
To bound , let be a nonempty moat in , for some . By the laminarity of , the patch contains at most one nonempty moat of . All the moats are pairwise disjoint, so by (4.2), (4.4) and the Cauchy–Schwarz inequality,
Using Lemma 3.3,
with equality only if , because is a simple triangulation of the sphere, and as such has no vertex of degree . Summing over all moats gives the inequality
(4.6) 
with equality only if .
The moats are disjoint, so by inequalitites (4.3), (4.5) and (4.6),
Hence, by the Cauchy–Schwarz inequality and (4.1),
(4.7)  
(4.8)  
This completes the proof of the first part of Theorem 4.2.
To prove the inequality , it suffices to observe that by Lemma 2.1. Now suppose that . By Lemma 4.1, there exists a packing of ,  and moats in of total width . Then , i.e., all vertices of degree less than have odd degree, namely, or . Equality holds in (4.6) and in (4.8), so . Furthermore, equality holds in (4.7), so there is a natural number such that for every . Therefore, every has degree , so . By Lemma 3.3 each moat has area , so . Hence, is the union of twelve facedisjoint moats , for (see Figure 4.1). Each can be identified with a face of a regular dodecahedron, which shows that contains a subgroup isomorphic to . On the other hand, the dual graph of is a fullerene graph, and it can be shown (see e.g. [7]) that the largest possible automorphism group of a fullerene graph is isomorphic to . Hence, .
Conversely, suppose is a triangulation of the sphere with , all vertices of degree and , and . Then it can be shown (see [4, 10]) that can be constructed by pasting triangular regions of the (infinite) regular triangulation of the plane into the faces of a regular icosahedron (this is sometimes known in the literature as the Goldberg–Coxeter construction). The construction is uniquely determined by a dimensional vector , known as the Goldberg–Coxeter vector (see Figure 4.2). Since , we must have or . The area of is given by the formula . The condition implies that the Goldberg–Coxeter vector of is , which means that the distance between any pair of vertices in is at least . Therefore, is a packing of moats of total width , so . ∎
5. Proof of Theorem 1.1
Given a triangulation of the sphere, we construct the refinement as follows. First, we subdivide each edge of , that is, we replace it by an internally disjoint path of length , and then we add three new edges inside every face, incident to the three vertices of degree . (For an illustration, see Figure 5.1.) Therefore, every face of is divided into four faces of . Observe that all the vertices in have degree in , so if is the set of odddegree vertices of , then is also the set of odddegree vertices of .
Lemma 5.1.
If is a triangulation of the sphere and is a subset of even cardinality, then .
Theorem 4.2 and Lemma 5.1 immediately give the following tight upper bound on the minimum size of a postman set in a plane triangulation with maximum degree .
Theorem 5.2.
Let be a triangulation of the sphere with faces and maximum degree at most . If is the set of odddegree vertices of , then
In particular, always holds, with equality if and only if all vertices have degree and , for some , and .
Proof.
Let be a triangulation of the sphere with maximum degree at most , and let be its refinement; observe that . By Lemma 5.1 and Theorem 4.2,
as required.
If , then , so by the second part of Theorem 4.2, all vertices in have degree and , and this must clearly hold in . Furthermore, , for some , so . Since is even, , for some , so . We also have .
Conversely, suppose is a triangulation of the sphere with , all vertices of degree and , and . By Theorem 4.2 , so . ∎
Proof of Theorem 1.1.
Let be a connected cubic plane graph on vertices with all faces of size at most , with