A bound on the number of perfect matchings in Klee-graphs †

. The famous conjecture of Lov´asz and Plummer, very recently proven by Esperet et al. (2011), asserts that every cubic bridgeless graph has exponentially many perfect matchings. In this paper we improve the bound of Esperet et al. for a speciﬁc subclass of cubic bridgeless graphs called the Klee-graphs . We show that every Klee-graph with n ≥ 8 vertices has at least 3 · 2 ( n +12) / 60 perfect matchings.


Introduction
In this paper, we focus on a specific class of planar cubic bridgeless graphs, namely the Klee-graphs, defined as follows.A graph G is a Klee-graph if G = K 4 or there exists a Klee-graph G such that G can be obtained from G by replacing a vertex by a triangle (see Figure 1).For a given undirected graph G, let V (G) be the set of its vertices and let E(G) be the set of its edges.A matching in a graph G is a set M ⊆ E(G) such that every vertex is an endvertex of at most one edge in M .A perfect matching is a matching where every vertex is an endvertex of exactly one edge in the matching.
Let us now mention a well-known conjecture of Lovász and Plummer [8,Conjecture 8.1.8]Conjecture 1.1 (Lovász-Plummer) Every bridgeless cubic graph G has exponentially many (as a function of |V (G)|) perfect matchings.
In 1982, Edmonds et al. [2] proved that every bridgeless cubic graph on n vertices has at least n/4 + 2 perfect matchings.This linear lower bound was improved by Král' et al. [7] to n/2 and then by Esperet et al. [5] to 3  4 n − 10.In 2009 Esperet et al. [4] came up with a deep and complex proof of a first known superlinear bound.
Meanwhile, the Lovász-Plummer conjecture has been proven for cubic bipartite graphs by Voorhoeve [10] and extended to all regular bipartite graphs by Shrijver [9].Recently Kardoš et al. [6] proved that fullerene graphs, i.e., planar cubic 3-connected graphs with only pentagonal and hexagonal faces, have a very large number of perfect matchings -their lower bound is 2 (n−380)/61 .A result by Chudnovsky and Seymour [1] from 2008 shows that the conjecture is true for all planar cubic bridgeless graphs.However, their exponential bound is very small, namely, they assert that every n-vertex planar cubic graph G has at least 2 n/655978752 perfect matchings.
The conjecture of Lovász and Plummer was finally proven in 2010 by Esperet et al. [3], who give a 2 n/3656 lower bound for the number of perfect matchings in an arbitrary n-vertex cubic bridgeless graph.
On the other hand, Klee-graphs play important role in the matchings and cubic graphs theory and therefore obtaining a better bound for the number of perfect matchings in Klee-graphs seems interesting.Every 3-edge-connected cubic graph that is not a Klee-graph is double-covered (i.e., every edge belongs to at least two perfect matchings) [5].Thus, one may expect that Klee-graphs are the 3-edge-connected cubic graphs with fewest perfect matchings.We start by introducing some notation in Section 2, then we prove a few basic lemmas in Section 3 and finally we do calculations and show that every Klee-graph with n ≥ 8 vertices has at least 3 • 2 (n+12)/60 perfect matchings in Section 4. In Section 5 we provide an upper bound for the number of perfect matchings in Klee-graphs.More precisely, we show an infinite family of Klee-graphs with at most c 2 n/17.285perfect matchings, where c is some absolute constant.

Notation and definitions
Let us start with some notation.For a given set The following lemma helps us fix a naming convention on the edges of a Klee-graph.Lemma 2.1 For any Klee-graph G, the edge set E(G) can be uniquely partitioned into three pairwise disjoint perfect matchings M 1 , M 2 and M 3 .In other words, any Klee-graph G is 3-edge-colorable, and the coloring is unique up to a permutation of the colors.
Proof: We prove it by induction on the number of vertices of G.The claim is obvious in the unique 4-vertex Klee-graph, K 4 .Now we focus on the Klee-graph construction step, i.e., we replace a vertex x in a graph G by a triangle a 1 a 2 a 3 obtaining a graph G.To prove the lemma, it is sufficient to show a bijection between partitions of E(G ) and E(G) into three perfect matchings. Assume 3 is a partition of E(G) into three perfect matchings (we assume a 4 = a 1 and a 5 = a 2 ).
In what follows we prove that the 3-edge coloring is unique up to a permutation of colors.For a partition of E(G) into three perfect matchings M 1 G , M 2 G , M 3 G , observe that the edges of the triangle a 1 a 2 a 3 belong to different sets M i G .Without loss of generality, assume that a i+1 a i+2 ∈ M i G for i = 1, 2, 3 (again, a 4 = a 1 and a 5 = a 2 ).We infer that 2 From this point, if we consider a Klee-graph G, by M 1 G , M 2 G and M 3 G we denote the unique (up to a permutation) partition of E(G) into three perfect matchings.
Let us now formalize the notation of the Klee-graph construction procedure, so we can later use arguments based on the below definition.We introduce a construction tree T of a Klee-graph G.It is an ordered rooted tree (i.e., for each node, its children are ordered) with subsets of V (G) as labels of the nodes.The root of T has four children, whereas all other nodes are either leaves or have three children.In other words, T is a 4-regular tree rooted at a non-leaf node.The labels of the leaves are singletons, the label of a node is the union of the labels of its children and the label of the root is V (G).We sometimes abuse the notation and identify the nodes with theirs labels.
Moreover, in the definition we maintain the following property: for every non-root node with label W in T the set ε G (W ) has three elements, and they belong to different sets in the partition of E(G) into three perfect matchings.We denote the edge of ε G (W ) that belongs to M i G by e i W .We denote also e i {x} by e i x for short.Inductively, we define the construction tree as follows: 1.A construction tree of K 4 with vertex set V (K 4 ) = {a, b, c, d} is a tree with its root labeled {a, b, c, d} and four leaves labeled {a}, {b}, {c} and {d}.If the leaves are ordered in this way, we assume that the edges are labeled as follows: e 2. Assume that a Klee-graph G is created from a Klee-graph G by replacing a vertex x ∈ V (G ) by a triangle a 1 a 2 a 3 .Let T be a construction tree of G and let e i x = xv i for i = 1, 2, 3. Assume that the vertex v i is connected to a i in G. Then the tree T is obtained from T as follows: • The leaf {x} in T is replaced by a node {a 1 , a 2 , a 3 } and with three new leaves {a 1 }, {a 2 }, {a 3 } connected in this order as its children.
• Every other label W T in T is replaced by a label W T defined as follows: We note that the edge labels behave as in the proof of Lemma 2.1.That is, for all i = 1, 2, 3 we have e i ai = e i {a1a2a3} = a i v i .Moreover e 3 a1 = e 3 a2 = a 1 a 2 , e 2 a1 = e 2 a3 = a 1 a 3 and e 1 a2 = e 1 a3 = a 2 a 3 .The edge numbering is depicted in Figure 3.A number i near an edge e = uv means that e = e i v = e i u ∈ M i G .We note also that for any non-root node W T in T we have the edges e i W T for i = 1, 2, 3 as follows: Let us now check if the definition of T really maintains the aforementioned property that for every non-root node with label W T in T , ε G (W T ) is a 3-edge-cut (i.e., a set of three edges, such that their removal disconnects the graph), and the edges belong to different perfect matchings in the partition by replacing all edges of the form xv i by a i v i and Note that, given a Klee-graph G, its construction tree is not defined uniquely.For instance the unique Klee-graph with 8 vertices (see Figure 2) has some freedom in choosing its construction tree: we can create this graph by replacing two vertices of K 4 with triangles -obtaining a construction tree of depth 2, or we can replace one vertex of K 4 with a triangle and then replace one of the new vertices with another triangle -obtaining a construction tree of depth 3.
Fix a Klee-graph G and any its construction tree T .Let W be one of the non-root nodes in T and let v i be the endvertex of e i W that is not contained in W (i = 1, 2, 3).The subgraph G[W ] is called a tripod and the edges ε We extend the definition of a construction tree to tripods: for a tripod W its construction tree, denoted as T (W ), is the subtree of T rooted at W .Note that if T (W ) has k non-leaf nodes, then it has 2k + 1 leaves and |W | = 2k + 1, so the tripod size is always an odd integer.Note that if a tripod W is not a single vertex, it consists of three smaller tripods, namely the children W 1 , W 2 , W 3 of W in the construction tree.It is straightforward from the definition of the construction tree that the legs of tripods are enumerated as in Figure 5. Later on, when we consider a tripod in a Klee-graph G, we implicitly assume that we are given a fixed construction tree for G.

Klee-graphs structure
In this section we gather a few structural results about Klee-graphs.Lemma 3.1 Let G be a Klee-graph with a construction tree T and let W ⊆ V (G) be a tripod in G. Then the tripod graph G W is a Klee-graph, too.
Proof: It is sufficient to build a construction tree T for the tripod graph G W , which can be obtained by attaching to the root node the subtree T (W ) and three leaves with labels containing the vertices of the triangle added to W in the tripod graph G W . 2 Regarding the triangles in the Klee-graphs, one can easily observe the following: Lemma 3.2 Let G be a Klee-graph on at least 6 vertices.Then, G has at least two triangles and all triangles are vertex-disjoint.
Proof: We prove it by induction on the number of vertices of G.The unique 6-vertex Klee-graph has two vertex-disjoint triangles, see Figure 2. Now we focus on the Klee-graph construction step, i.e., we replace a vertex x in a graph G by a triangle abc obtaining a graph G .By the induction hypothesis, in G there exists at most one triangle containing x.In G we create a new triangle abc and destroy at most one triangle from G -the one containing x.Notice that there is no triangle with any of the vertices a, b, c in G except for abc, so the new triangle is vertex-disjoint from the other ones in G .Therefore the construction step cannot decrease the number of triangles in the graph. 2 The next easy lemma ensures that if we collapse a triangle into a vertex, we are still in the class of Klee-graphs.Lemma 3.3 Let G be a Klee-graph on at least 6 vertices and let abc be a triangle in G. Let G be the graph obtained from G by contracting vertices a, b, c into a vertex x and removing the loops.Then, G is a Klee-graph too.
Proof: We prove it by induction on the number of vertices of G.If we contract one triangle in the unique 6-vertex Klee-graph we obtain K 4 .So assume G has at least 8 vertices.By the definition of the Klee-graphs, G can be obtained from Klee-graph G 0 by replacing a vertex y ∈ V (G 0 ) by a triangle pqr.If pqr = abc, we have G = G 0 (taking x = y) and we are done.Otherwise, by Lemma 3.2, {a, b, c} ∩ {p, q, r} = ∅ and the triangle abc exists in G 0 .Let G 0 be the graph constructed by contracting the triangle abc in G 0 into one vertex x.Then, by induction hypothesis, G 0 is a Klee-graph and G is obtained from G 0 by replacing y by a triangle pqr, so it is a Klee-graph too. 2 Lemma 3.4 Let G be a Klee-graph and let abc be a triangle in G. Then there exists a construction tree T for G such that the children of the root of T are {a}, {b}, {c} and V (G) \ {a, b, c}.In other words, there exists a construction tree such that G is a tripod-graph for the tripod V (G) \ {a, b, c}.
Proof: We prove by induction on |V (G)|.For G = K 4 , the claim is obvious.Take G with |V (G)| ≥ 6.By Lemma 3.2 there exists a triangle pqr disjoint from abc.By Lemma 3.3, the graph G constructed from G by contracting the triangle pqr into a vertex x is also a Klee-graph.By induction hypothesis, there exists a construction tree T for G such that the children of the root of T are {a}, {b}, {c} and V (G )\{a, b, c}.We replace the leaf {x} by the node {p, q, r} and its children {p}, {q}, {r}, as described in the definition of a construction tree.This way we obtain the desired tree T . 2

Counting perfect matchings
Let G be a Klee-graph with some fixed construction tree T and let W be a tripod in G. Since |W | is odd, for any perfect matching M in G precisely one or all three legs of W are in M .Denote by P i (W ) the number of perfect matchings in the tripod graph G W which use only leg e i W (i = 1, 2, 3), and denote by P (W ) the number of perfect matchings in G W which use all three legs.Let us define P(W ).A motivation for this product will be explained later.
Note that the following hold: Lemma 4.1 Let W be a tripod in a Klee-graph G, and G a graph constructed by extending one vertex x ∈ W into a triangle abc.Let W be the tripod in G that corresponds to W , i.e., W = W \ {x} ∪ {a, b, c}.Then P i (W ) ≤ P i (W ) for i = 1, 2, 3 and P (W ) ≤ P (W ).Moreover, P i (W ) ≥ 1 for i = 1, 2, 3.
Proof: Assume that N G ({x}) = {x a , x b , x c } and each vertex v ∈ {a, b, c} is connected to x v in G .Then any perfect matching M in G W can be extended to a perfect matching M in G W by replacing the edge xx v (v ∈ {a, b, c}) by the edge vx v and by the edge with both endvertices in {a, b, c} \ {v}.Finally note that the extensions of distinct matchings are distinct.
To see that P i (W ) ≥ 1 for i = 1, 2, 3 note that the part of the perfect matching Lemma 4.2 Let G be a Klee-graph with a tripod W of size greater than one.Let W 1 , W 2 and W 3 be the children of W in the construction tree.Then the following formulas hold: In particular, P(W ) ≥ P(W j ) for j = 1, 2, 3.
P 1 (W 1 ) Proof: In each equation, we consider two cases which are illustrated in Figure 6.By symmetry, it suffices to prove the formulas for P 1 (W ) and P (W ).Let us start with P 1 (W ).The leg e 1 W1 must be used and legs e 2 W2 and e 3 W3 must not.Recall that from every tripod we may use one or all three legs in any perfect matching.Therefore, if we use the leg e 1 W1 then we may use all legs of W 1 -in this case we use legs e 3 W2 and e 2 W3 -or we may use only leg e 1 W1 of W 1 and in this case we need to use the edge e 1 W2 = e 1

W3
between tripods W 2 and W 3 .To obtain the formula for P (W ), note that we may either use all of the edges between tripods W 1 , W 2 , W 3 or none of them, which proves the desired equations.Finally, since P i (W j ) ≥ 1 for i, j ∈ {1, 2, 3}, we have P 1 (W ) ≥ P 1 (W 1 ), P 2 (W ) ≥ P 2 (W 1 ), P 3 (W ) ≥ P 3 (W 1 ) and therefore P(W ) ≥ P(W 1 ).Similarly P(W ) ≥ P(W j ) for j = 2, 3. 2 Before we proceed to the main result, we need to do some calculations by hand to provide the basis for the inductive proof of the main theorem.Lemma 4.3 Let W be a tripod of size at least 5.Then, either W is one of the tripods depicted in Figure 7 or P(W ) ≥ 4.
Proof: Note first that the left tripod in Figure 7 is the unique tripod on 5 vertices, so we may assume |W | ≥ 7.
Let W 1 , W 2 and W 3 be the children of W in the construction tree.Without loss of generality assume that  Let us note that, unfortunately, it is not obvious how to use the sum P 1 (W ) + P 2 (W ) + P 3 (W ) in the following inductive proof since it is possible that in the equations from Lemma 4.2 big values will be multiplied by small values resulting in something too small to preserve the bound.Therefore we make use of the product P(W ) = P 1 (W ) • P 2 (W ) • P 3 (W ) in the main theorem of this section.Without loss of generality we may assume that W (1) = W (0) 1 = W 1 .Therefore: P 1 (W (0) ) ≥ P 1 (W (1) ) + P (W (1) ) P 2 (W (0) ) ≥ P 2 (W (1) ) P 3 (W (0) ) ≥ P 3 (W (1) ) P (W (0) ) ≥ P 1 (W (1) ).

Fig. 1 :
Fig. 1: Replacing a vertex by a triangle in a cubic graph.
be the set of edges connecting D with V (G) \ D, and let N G (D) ⊆ V (G) \ D be the set of neighbors of D. Denote by G[D] the subgraph induced by D.

Fig. 3 :
Fig. 3: Labeling edges after replacing a vertex by a triangle.A number i near an edge e = uv means that e = e i v = e i u ∈ M i G .

Fig. 4 :
Fig. 4: Tripod graph GW .A number i near an edge e means that e = e i W ∈ M i G .

Fig. 5 :
Fig. 5: Labeling of legs of a tripod and its children.A number i near an edge e that has an endpoint in a tripod Y ∈ {W, W1, W2, W3} means that e = e i Y ∈ M i G .