Connected cubic graphs with the maximum number of perfect matchings
aa r X i v : . [ m a t h . C O ] J un Connected cubic graphs with the maximumnumber of perfect matchings
Peter HorakUniversity of Washington, Tacoma [email protected]
Dongryul KimStanford University [email protected]
June 25, 2020
Abstract
It is proved that for n ≥
6, the number of perfect matchings in a simpleconnected cubic graph on 2 n vertices is at most 4 f n − , with f n being the n -th Fibonacci number. The unique extremal graph is characterized aswell.In addition, it is shown that the number of perfect matchings in anycubic graph G equals the expected value of a random variable defined onall 2-colorings of edges of G . Finally, an improved lower bound on themaximum number of cycles in a cubic graph is provided. Contents
It is natural to ask how many cycles, Hamiltonian cycles, and 2-factors a graphcan have. To the best of our knowledge this question for cycles was considered1or the first time by Ahrens [1] in 1897. This paper focuses on the number of2-factors in cubic graphs.We first note that in a cubic graph, the complement of a 1-factor (i.e., aperfect matching) is a 2-factor, and vice versa. Therefore, for a cubic graph G , the number of perfect matchings, the number of 1-factors, the number of2-factors are all equal, and we denote this byPerMat( G ) = Fac( G ) . There is an extensive literature on the number of perfect matchings in cubicgraphs. We mention here only the most pertinent results. It is shown in [6]that the number of perfect matchings in a 2-connected cubic graph is at leastexponential with its order, thus confirming an old conjecture of Lov´asz andPlummer [9].As for the maximum number of perfect matchings, Alon and Friedland [3]proved a general result. Its restriction to cubic graphs states:
Theorem 1.1 (Alon–Freidland [3], 2008) . For a simple cubic graph G on n vertices, PerMat( G ) ≤ n/ . This bound is tight, and it is attained by taking the disjoint union of bipartitecomplete graphs K , . In other words, the above theorem says that the complete bipartite graph K , has the highest “density” of perfect matchings among all cubic graphs;thus the disjoint union of its copies constitutes the extremal graph. However,this result does not provide any insight into the structure of extremal connectedcubic graphs. Remark 1.2.
When G is bipartite, Theorem 1.1 is an immediate consequence ofthe Br`egman–Minc inequality, proved by Br`egman [4] in 1973. The contributionof Alon and Friedland was introducing a clever trick that extended the resultfrom bipartite graphs to all graphs. We later use the same trick, in the form ofLemma 3.6.If only connected cubic (but not necessarily simple) graphs are consideredthen Galbiati [7] proved the following theorem. Theorem 1.3 (Galbati [7], 1981) . For a connected cubic (multi) graph G on n vertices, PerMat( G ) ≤ n + 1 . This bound is tight, and it is attained by taking a cycle of length n and puttingparallel edges alternatively. The main result of this paper characterizes the extremal graphs if only simpleconnected cubic graphs are taken into account. It is somewhat counter-intuitivethat the extremal graphs M n are 2-connected but not 3-connected for large n .2 heorem 1.4. Let G be a simple, connected cubic graph on n vertices. Then, PerMat( G ) ≤ m n , and moreover, equality is achieved if and only if G is iso-morphic to the graph M n (see Section 2 for the definition of m n and M n ). Inparticular, for n ≥ we have a sharp bound PerMat( G ) ≤ f n − , where f n − is the ( n − th Fibonacci number. In addition, we derive a formula for counting the number of perfect match-ings.
Theorem 1.5.
For a cubic graph G of order n , the number of perfect matchingscan be calculated as the expected value PerMat( G ) = E( X ) , where X is a randomvariable defined on the set of all -colorings c on the edges of G , each coloringequally likely, and X ( c ) = ( − m c , where m c is the number of vertices of G incident in c with three edges of the same color. The above formula is not feasible for practical calculations but we believe itis of a theoretical value. The proof is based on interpreting the number of perfectmatchings as an evaluation of a suitable quantum field theory. We include onlya sketch of the proof in Section 4.Finally, we turn to the original question of Ahrens [1], on the maximumnumber of cycles in a connected graph G = ( V, E ). It is most convenient tostudy the number of cycles in a connected graph G with respect to its cyclomaticnumber r ( G ) = | E | − | V | + 1, since the number of cycles is bounded by 2 r ( G ) −
1. Let Ψ( r ) be the maximum number of cycles among all graphs with thecyclomatic number equal to r . Entringer and Slater [5] showed that the problemof determining Ψ( r ) can be reduced to cubic graphs; they proved that, for r ≥ r and Ψ( r ) cycles.In addition they conjectured that Ψ( r ) ∼ r − . So far the best upper bound hasbeen provided by Aldred and Thomassen [2] who proved there that Ψ( r ) ≤ r .As for the lower bounds on Ψ( r ), the first lower bound Ψ( r ) ≥ r − + f ( r ),where the error function f ( r ) is exponential has been given in [8]. In Section 5,we show that a graph obtained from C n × K by replacing consecutive pairs of“parallel” rungs by “crossing” ones provides a bound on Ψ( r ) with an improvedexponential error term. Theorem 1.6.
There exists a constant c > for which Ψ( r ) ≥ r − + cr r/ for all sufficiently large r . For a more precise bound, we refer the reader to Section 5.
Before we prove Theorem 1.4, we collect properties that we will later use toprove Theorem 1.4. The results in this section mostly follow from a routineverification, hence we leave some of the details to the reader.3e define the Fibonacci numbers by f n = 1 √ ϕ n − ϕ − n ) , ϕ = 1 + √ , so that they satisfy f = 0 , f = 1 , f = 1 , f n = f n − + f n − for all integers n .For the purpose of stating and proving Theorem 1.4, we define m n for n ≥ m = 3 , m = 6 , m = 9 , m = 13 , m n = 4 f n − for n ≥ . We collect the inequalities among m n that we will use later. Note that weasymptotically have m n ∼ cϕ n where c ≈ .
106 and ϕ ≈ .
618 is the goldenratio; this is useful for doing a quick sanity check.
Lemma 2.1. (i) For n ≥ , we have m n = m n − + m n − .(ii) For n ≥ , we have m n − < m n .(iii) For n ≥ , we have m n − < m n .(iv) For n ≥ , we have m n − ≤ m n , with equality only for n = 8 .(v) For n ≥ , we have m n − < m n .(vi) For n ≥ , we have m n +1 < √ m n .(vii) For n ≥ , we have m n < m n .(viii) For a, b ≥ , if a + b ≥ then m a m b < m a + b +1 .Proof. (i) follows from the definition that m n = 4 f n − for n ≥
6. For (ii),we can check the inequality by hand when 6 ≤ n ≤
8. When n ≥
9, we useinduction; assuming that the inequality holds for smaller n ≥
6, we can write32 m n − = 32 m n − + 32 m n − < m n − + m n − = m n by (i), since n ≥
9. The other inequalities (iii), (iv), (v), (vi) can be provedusing a similar inductive argument.For (vii), we first do the case n ≤ n ≥
6, we note that m n = 4 f n − = 4( f n + f n − ) = 14 ( m n +1 + m n ) <
14 (( √ m n ) + m n ) = m n by (vi). Here, the identity f n − = f n + f n − is folklore.For (viii), we induct on a + b . When max( a, b ) ≤
7, we can check manually.If max( a, b ) ≥
8, without loss of generally assume that a ≥
8. Then by (i) andthe inductive hypothesis, m a m b = m a − m b + m a − m b < m a + b + m a + b − = m a + b +1 . Here, the induction hypothesis applies because ( a −
2) + b ≥ − M · · ·· · · Figure 2: The graph M n for n ≥ Definition 2.2.
For n ≥
2, we define the simple connected cubic graph M n asfollows: • M = K ; • M = K , ; • M is K , with a perfect matching removed; • M is the M¨obius ladder on 10 vertices, see Figure 1; • for n ≥ M n is the ladder graph with K , inserted at both ends, seeFigure 2. Proposition 2.3.
For all n ≥ , we have Fac( M n ) = m n .Proof. For n ≤
7, we can count the number of 2-factors explicitly, and checkthat the number of 2-factors of M n is exactly m n . For n ≥
8, we induct on n .It is shown in Section 3, proof of Theorem 1.4, Subcase 1-1, thatFac( M n ) = Fac( M n − ) + Fac( M n − ) . Then it follows from Lemma 2.1, (i) that Fac( M n ) = m n − + m n − = m n bythe inductive hypothesis. 5 Proof of Theorem 1.4
For the reader’s convenience, the proof of Theorem 1.4 will be presented interms of 2-factors instead of perfect matchings. We recall that the complementof a perfect matching in a cubic graph is 2-factor.The general strategy for proving Theorem 1.4 is to first prove the theoremfor bipartite graphs. Then using a trick of Alon and Friedland [3], we deducethe general case. We start with an auxiliary result.
Lemma 3.1.
Let G be a simple connected bipartite cubic graph.(i) The graph G does not have a bridge.(ii) For every edge xy of G , the induced subgraph on V ( G ) − { x, y } has atmost two components.Proof. (i) If it has a bridge, we may remove the edge and take a component ofthe resulting graph. This is a bipartite graph with one vertex of degree 2 andall other vertices having degree 3. Counting the number of edges, we see thatit is both a multiple of 3 and equal to 2 modulo 3, arriving at a contradiction.(ii) Denote by G ′ the induced subgraph on V ( G ) \ { x, y } and assume that G ′ has at least three components. Let the neighbors of x be y, a, b and theneighbors of y be x, c, d . Because G was connected, each one of the componentsof G ′ contains at least one of the four vertices a, b, c, d . By part (i), the edge xy is not a bridge in G . This means that there is a path in G ′ connecting either a or b to either c or d . Hence, without loss of generality, we may as well assumethat a, c are in the same component in G ′ . Then b is in a different componentthan a, c, d in G ′ , which means that the edge xb must be a bridge in G . Thiscontradicts part (i). Definition 3.2.
For a simple connected bipartite cubic graph G , we say thatan edge e = uv is a ladder-bridge if the induced subgraph on V ( G ) \ { u, v } isdisconnected, or equivalently, has two components.We first prove Theorem 1.4 for bipartite graphs. Theorem 3.3.
Let G be a simple connected cubic bipartite graph on n vertices(so automatically n ≥ ). Then PerMat( G ) = Fac( G ) ≤ m n , and moreover,equality is achieved if and only if G is isomorphic to the graph M n . Remark 3.4.
For n ≤
8, we content ourselves with using a computer to verifythe statement. (There are 60 simple connected bipartite cubic graphs with atmost 16 vertices in total.) However, it is possible to avoid using a computerat all. We can get away with verifying the statement by hand for n ≤ n = 7 , yac bdA B Figure 3: Case 1—a connected bipartite cubic graph G with ladder-bridge xyxyac bd efA B Figure 4: Subcase 1-1—when xy is next to another ladder-bridge bd Proof of Theorem 3.3.
We use induction on n . For n ≤
8, we can perform acomputer search on all the cubic bipartite graphs. Hence, let us assume that n ≥ n . Case 1.
The graph G has a ladder-bridge xy . Let us write V ( G ) \ { x, y } = A ∪ B , where A and B are the components. Note that if all neighbors of x arein A ∪ { y } , then one of the edges adjacent to y is a bridge of G , contradictingLemma 3.1. Hence one of the neighbors of x is in A , one of them is in B , andsimilarly for y . Define a, c ∈ A and b, d ∈ B so that the neighbors of x are y, a, b and the neighbors of y are x, c, d , as in Figure 3. Subcase 1-1.
Suppose either a and c are connected by an edge or b and d are connected by an edge. Without loss of generality, we assume that b, d areconnected by an edge. Let the neighbors of b be x, d, e , and the neighbors of d be y, b, f , as in Figure 4. We consider two other graphs: G ′ , which is the graphobtained by removing the vertices x, y and connecting ad, cb , and G ′′ , whichis the graph obtained by removing vertices x, y, b, d and connecting ae, cf , seeFigure 5. Note that both G ′ , G ′′ are again simple connected bipartite cubicgraphs, where connectivity of G ′′ follows from the fact that there is a path in A connecting a, c . Any 2-factor of G will either contain both ax, cy or containneither, because these are the only edges connecting a vertex in A and a vertexnot in A . Using this, we see that there are five possible shapes a 2-factor of G can take on the induced subgraph on { a, b, c, d, e, f, x, y } , listed on the leftmostcolumn of Figure 5. Similarly, we can list all the possible shapes a 2-factor of G ′ or G ′′ can have on the induced subgraph of { a, b, c, d, e, f } or { a, c, e, f } , andthese are depicted on the rightmost column of Figure 5.We now modify the 2-factor of G to either a 2-factor of G ′ or a 2-factor of G ′′ , by simply removing all the edges between a, b, c, d, e, f, x, y and filling inthat part with an appropriate diagram. If we do this process by taking the i thdiagram on the leftmost column of Figure 5 replacing with the i th diagram onthe rightmost column of Figure 5. The process is reversible, as we can similarly7 G ′ G ′′ Figure 5: Subcase 1-1—modifying the graph G to G ′ and G ′′ take the right hand side diagram and replace it with the left hand side diagram.This shows that Fac( G ) = Fac( G ′ ) + Fac( G ′′ ) . The inductive hypothesis applies to both G ′ and G ′′ , as they are simple con-nected bipartite cubic graphs with number of vertices less than 2 n . ThereforeFac( G ) = Fac( G ′ ) + Fac( G ′′ ) ≤ m n − + m n − = m n , where the last equality holds by Lemma 2.1, (i), since n ≥ G ′ is isomorphic to the graph M n − and G ′′ is isomorphic to thegraph M n − . In G ′ , the edge bd is a ladder-bridge. To recover G from G ′ , weneed to insert another ladder-bridge, and it is easy to verify that the resultinggraph is always isomorphic to M n . Subcase 1-2.
Suppose now that there is no edge between a, c and alsobetween b, d . This time, we modify the graph G to G ′ by removing the vertices x, y and then connecting a, c and b, d , see Figure 6. Then G ′ is a simple bipartitecubic graph, even though it is not connected. Using the same process of replacingthe i th configuraiton of the leftmost column with the i th configuration of therightmost column, from each 2-factor of G we get a 2-factor of G ′ . Moreover, it isclear that distinct 2-factors of G give distinct 2-factors of G ′ , even though some2-factors of G ′ do not appear by this process. Therefore Fac( G ) ≤ Fac( G ′ ). Onthe other hand, G ′ as two components, say G ′ = G ′ ∪ G ′ , where both G ′ , G ′ aresimple connected bipartite cubic graphs. Then the inductive hypothesis applies,so Fac( G ) ≤ Fac( G ′ ) = Fac( G ′ ) Fac( G ′ ) ≤ m a m n − a − where G ′ has 2 a vertices and G ′ has 2( n − a −
1) vertices. Because n ≥
9, fromLemma 2.1, (viii), we obtainFac( G ) ≤ m a m n − a − < m n . G ′ Figure 6: Subcase 1-2—modifying the graph G to G ′ x yzwa bcd Figure 7: Case 2—a cubic bipartite G with a 4-cycle xyzw Case 2.
Now suppose there exists a 4-cycle xyzw in G . Denote by a, y, w the neighbors of x , by b, x, z the neighbors of y , by c, y, w the neighbors of z ,and by d, x, z the neighbors of w , see Figure 7. By definition, x, y, z, w are alldistinct points, but there is no reason for a, b, c, d to be all distinct. It is possiblethat a = c or b = d or both. Subcase 2-1.
First consider the case when a = c and b = d . Now thevertices a, b, x, y, z, w are all distinct. Moreover, a cannot be connected to b by an edge, otherwise we would have V ( G ) = { a, b, x, y, z, w } and G = K , ,contradicting our assumption that n ≥
9. Denote by x, z, e the neighbors of a and y, w, f the neighbors of f , see Figure 8. As a and b are not neighbors, wesee that all the vertices a, b, e, f, x, y, z, w are distinct. If e and f are connected,then ef becomes a ladder-bridge of G , hence this case is already covered inCase 1. Therefore we assume without loss of generality that e and f are notconnected by an edge.Given such a graph G , we define a new graph G ′ by removing the vertices a, b, x, y, z, w and then connecting e and f . The resulting graph is simple, as e and f were not already connected, and also connected cubic bipartite. Weagain list the possible configurations of 2-factors of G restricted to this portion,of which there are 6 as listed on the leftmost column of Figure 9. We mayagain use the process of replacing the configurations of the leftmost column byconfigurations of the rightmost column. This time, the process is not injective,9 yz wa be f Figure 8: Subcase 2-1—when a = c and b = dG G ′ Figure 9: Subcase 2-1—modifying the graph G to G ′ but every 2-factor of G ′ can occur in at most 4 different ways. This shows thatFac( G ) ≤ G ′ ) ≤ m n − < m n by Lemma 2.1, (iv). Subcase 2-2.
Now suppose that only one of the equalities from a = c and b = d hold. Without loss of generality, assume that a = c and b = d . Denoteby x, z, e the neighbors of a , see Figure 10. It is clear that a, x, y, z, w are alldistinct, and also distinct from b, d, e . The only possible equalities between thepoints a, b, d, e, x, y, z, w are d = e or b = e . But if any of these equalities holds,we are reduced to Subcase 2-1. For instance, if b = e then axyz becomes a4-cycle satisfying the assumptions of Subcase 2-1. Therefore we may as wellassume that all the vertices a, b, d, e, x, y, z, w are distinct.We now construct a new graph G ′ by deleting the vertices x, y, z, w and thenconnecting a with b and d . As b, d, e are distinct points, the resulting graph issimple. That is, G ′ is a simple connected bipartite cubic graph, hence satisfiesthe induction hypothesis. We list the possible 2-factors of G and consider thereplacing process as given by Figure 11. We observe that each 2-factor of G ′ can occur in exactly 2 ways, and thereforeFac( G ) = 2 Fac( G ′ ) ≤ m n − < m n by Lemma 2.1, (iii). 10 zy waeb d Figure 10: Subcase 2-2—when a = cG G ′ Figure 11: Subcase 2-2—modifying the graph G to G ′ Subcase 2-3.
We are now left with the case when a = c and b = d . Inthis Subcase, assume that { ab, cd } ∩ E ( G ) = ∅ and { bc, ad } ∩ E ( G ) = ∅ . Thenwithout loss of generality, we can assume that a is connected to both b and d by edges. Denote by e, a, y the neighbors of b and by f, a, w the neighborsof d . The vertices c, e, f cannot be all equal, because then we would have V ( G ) = { a, b, c, d, x, y, z, w } which contradicts n ≥
9. Thus either the threevertices c, e, f are all distinct, or two of them are equal and distinct from thelast one.
Subsubcase 2-3-1.
We first assume that two of c, e, f are equal, but notall of them coincide. Here, by symmetry of Figure 12, we may as well assumethat e = f but c = e . Let us denote by g, b, d the neighbors of e = f . Then allthe 10 vertices a, b, c, d, e, g, x, y, z, w are distinct. We now use the exact samestrategy as Subcase 2-1. If g and c are connected by an edge, then the edge cg becomes a ladder-bridge of G , hence we can deal with it using Case 1. If g and c are not connected by an edge, consider the graph G ′ obtained from G bydeleting the vertices a, b, d, e, x, y, z, w and connecting c and g by an edge. Then G ′ is a simple connected bipartite cubic graph, hence the inductive hypothesis11 y zwa b cd ef Figure 12: Subcase 2-3—when a = c and b = d but a is connected to b, dG G ′ Figure 13: Subsubcase 2-3-1—modifying the graph G to G ′ applies. From Figure 13, we see thatFac( G ) ≤ G ′ ) ≤ m n − < m n by Lemma 2.1, (v). Subsubcase 2-3-2.
We now suppose that c, e, f are all distinct. This timewe follow Subcase 2-2. Consider the graph G ′ obtained by deleting the vertices a, d, x, y, z, w from G and the connecting b to both c, f . Since c, e, f are distinctpoints, the modified graph G ′ is simple, and also connected bipartite cubic. Wecan list the possible 2-factors of G restricted to the subgraph as in Figure 14.Now every 2-factor of G ′ can be obtained from a 2-factor of G in exactly 3 ways,hence Fac( G ) = 3 Fac( G ′ ) ≤ m n − < m n − < m n by Lemma 2.1, (iv). Subcase 2-4.
Finally, we assume that a = c , b = d , and either { ab, cd } ∩ E ( G ) = ∅ , or { bc, ad } ∩ E ( G ) = ∅ . Without loss of generality, suppose that a, b are not connected by an edge and c, d are also not connected by an edge.In this case, as in Subcase 1-1, we consider two graphs. Let G ′ be the graphobtained from G by removing x, w and then connecting a, z and y, d . Let G ′′ G ′ Figure 14: Subsubcase 2-3-2—modifying the graph G to G ′ be the graph obtained from G by removing x, y, z, w and then connecting a, b and c, d . Since we have assumed that a, b and c, d are not already connected in G , we see that G ′ , G ′′ are both simple graphs. It is clear that G ′ is connected.We may assume that G ′′ is also connected, because if it is not connected thenremoving x, y from G disconnects the graph. This would mean that xy is aladder-bridge of G , but then Case 1 handles the situation. Therefore we maysuppose both G ′ and G ′′ are simple connected bipartite cubic graphs, and theinductive hypothesis applies. We now enumerate the posssible 2-factors and dothe replacement procedure according to Figure 15. From the usual analysis, itfollows that Fac( G ) ≤ Fac( G ′ ) + Fac( G ′′ ) ≤ m n − + m n − = m n by Lemma 2.1, (i).We now analyze the equality case. Similarly to Subcase 1-1, by the inductivehypothesis, equality holds only if G ′ and G ′′ are isomorphic to M n − and M n − .For the equality Fac( G ) = Fac( G ′ ) + Fac( G ′′ ) to holds, we further need thatthere is no 2-factor of G ′ using the edges az, zy, yb and also no 2-factor using theedges dy, yz, zc . By inspection, we see that the only edges e in M n − satisfyingthe above property for yz are precisely the ladder-bridges. This shows that yz corresponds to a ladder-bridge in M n − , and modifying G ′ to G shows that G is isomorphic to the graph M n . Case 3.
Since Case 2 was when G has a 4-cycle, we now assume that G has no 4-cycles. Since G is bipartite, this implies that all cycles of G has lengthat least 6. Pick an arbitrary vertex x , denote its neighbors by y, z, w , denotethe neighbors of y by x, a, b , the neighbors of z by x, c, d , and the neighbors of13 G ′ G ′′ Figure 15: Subcase 2-4—modifying the graph G to G ′ and G ′′ xy zwa b cdef Figure 16: Case 3—when there is no 4-cycle w by x, e, f , as in Figure 16. As G has no cycles of length smaller than 6, weimmediately see that the vertices x, y, z, w, a, b, c, d, e, f are all distinct.We now consider six graphs constructed from G . The graph G yj for j = 1 , G by removing the vertices x, y , and then adding the edges aw and bz for j = 1, and adding az and bw for j = 2. The other graphs G y , G y , G z , G z are constructed similarly, as depicted in Figure 17. It is clearthat the graphs G yj , G zj , G wj are all simple bipartite cubic graphs. We claimthat we may assume that these graphs are also connected for 1 ≤ i ≤
6. Forinstance, suppose that G y is not connected. Then G y with the edges aw and bz removed is also not connected. On the other hand, this graph is what we getwhen we remove the two vertices x, y from G . Thus G y not being connectedimplies that xy is a ladder-bridge in G . Since this is already covered in Case 1,we may as well assume that G y is connected. A similar argument works forall other graphs, hence we may assume that the modified graphs are all simpleconnected bipartite cubic graphs. Hence the inductive hypothesis holds for every G yj , G zj , G wj .Now we claim that given any 2-factor of G , it can be modified to a 2-factorof four of the graphs G vj (with v ∈ { y, z, w } ), in such a way that the 2-factorof G can be recovered from any of the modified 2-factors. By symmetry, weonly need to consider the case when the 2-factor on the induced subgraph on14 w G w G y G y G z G z Figure 17: Case 3—modifying the graph G to G yj , G zj , G wj { x, y, z, w, a, b, c, d, e, f } is as in Figure 18. In this case, the 2-factor of G can beturned into 2-factors of G z , G w , G y , G w without changing it outside the depictedregion. This shows that4 Fac( G ) ≤ X v,j Fac( G vj ) ≤ m n − and by Lemma 2.1, (ii), we have 6 m n − < m n . This concludes the proof.Let us now deduce Theorem 1.4 from Theorem 3.3. What we need is a cleverlemma of Alon and Friedland [3]. Definition 3.5.
Let G be a simple cubic graph. Define a new graph D ( G ) withvertices V ( D ( G )) = V ( G ) × { , } and edges E ( D ( G )) = { ( v, w,
2) : vw ∈ E ( G ) } . Note that D ( G ) is always a simple cubic bipartite graph if G is a simple cubicgraph. Moreover, if G is connected and not bipartite, then D ( G ) is connected. Lemma 3.6 (Alon–Friedland [3]) . Let G be a simple cubic graph. Then Fac( G ) ≤ Fac( D ( G )) . We can finally prove Theorem 1.4.
Proof of Theorem 1.4.
For n = 2, there is only one simple cubic graph on 4vertices, namely K . Hence there is nothing to prove.15 G w G z G w G y Figure 18: Case 3—modifying a 2-factor of G to 2-factors of G z , G w , G y , G w We now assume that n ≥
3. If G is not bipartite, then D ( G ) is connectedand bipartite, so Fac( G ) ≤ Fac( D ( G )) ≤ m n < m n by Lemma 3.6, Theorem 3.3, and Lemma 2.1, (vii). This shows that Fac( G ) 3, we define the cubic graph M C k as follows. There are 2 k vertices V ( M C k ) = { x i , y i : 1 ≤ i ≤ k } . Starting with these vertices, we add the edges x i x i +1 and y i y i +1 for 1 ≤ i ≤ k ,so that there are two k -cycles. Here, the indices are taken modulo k ; x i + k = x i and y i + k = y i . If k is even, add the edges (which we colloquially call “rungs”) { x i − y i , x i y i − : 1 ≤ i ≤ k/ } , and if k is odd, add the rungs { x i − y i , x i y i − : 1 ≤ i ≤ ( k − / } ∪ { x k y k } . It is then clear that M C k is always a simple connected cubic graph, with cyclo-matic number is given by r ( M C k ) = k + 1. See Figure 19 for a picture of M k for even k . Theorem 5.1. For k ≥ , the number of cycles in M C k is ( k + ( k + )2 k − k if k is even , k + ( k + )2 k − − (3 k + 5) if k is odd . In particular, for r ≥ , Ψ( r ) ≥ r − + ( r − )2 r − − ( r − for r odd , Ψ( r ) ≥ r − + ( r + )2 r − − (3 r + 2) for r even . x x x x k − x k y y y y y k − y k Figure 19: The graph M C k for even k Proof. We only prove the statement for k even, and leave the proof for k oddto the reader. Given a cycle C , we first observe that the parity of w i = C ∩ { x i x i +1 , y i y i +1 } )is independent of 1 ≤ i ≤ k/ C such that w i are all odd. Then either x i x i +1 is in C and y i y i +1 is not in C , or x i x i +1 is not in C and y i y i +1 isin C . There are in total 2 k/ ways to make this choice. For each choice, thereare again 2 k/ ways of completing the set C ∩ { x i x i +1 , y i y i +1 : 1 ≤ i ≤ k/ } to all of C . This shows that the total number of cycles C with w i odd is( C with w i odd) = 2 k/ · k/ = 2 k . We now count the number of cycles C such that w i are all even. In this case,either w i = 0 or w i = 2. We observe that the i such that w i = 2 must form aninterval modulo n . More precisely, there exist 1 ≤ s ≤ n and 0 ≤ d ≤ n suchthat w s = w s +1 = · · · = w s + d − = 1 , w s + d = w s + d +1 = · · · = w s + n − = 0 . When d = 0, the number of ways of completing to a cycle C is k/ 2, sincethe possible cycles are the 4-cycles x i − x i y i − y i . If 1 ≤ d ≤ k/ − 1, thenfor each choice of s there are 2 d +1 ways of completing it to a cycle. If d = k/ x i − x i and y i − y i are in C or x i − y i and y i − x i are in C . For C to be a 2 k -cycle instead of two k -cycles, the numberof x i − y i in C must be odd, hence there are 2 k − cycles. Then( C with w i even) = k n − X d =1 n · d +1 + 2 k − = k · (2 k +1 − 3) + 2 k − Hence, in aggregate, the number of cycles in M C k is2 k + (cid:16) k (cid:17) k +1 − k. The stated lower bound on Ψ( r ) follows directly from r ( M C k ) = k + 1.19 cknowledgment The authors are grateful to Noga Alon for discussions on the main result of thispaper. References [1] W. Ahrens. ¨Uber das Gleichungssystem einer Kirchhoff’schen galvanischenStromverzweigung. Math. Ann. , 49(2):311–324, 1897.[2] R. E. L. Aldred and Carsten Thomassen. On the maximum number of cyclesin a planar graph. J. Graph Theory , 57(3):255–264, 2008.[3] Noga Alon and Shmuel Friedland. The maximum number of perfect match-ings in graphs with a given degree sequence. Electron. J. Combin. , 15(1):Note13, 2, 2008.[4] L. M. Br`egman. Certain properties of nonnegative matrices and their per-manents. Dokl. Akad. Nauk SSSR , 211:27–30, 1973.[5] R. C. Entringer and P. J. Slater. On the maximum number of cycles in agraph. Ars Combin. , 11:289–294, 1981.[6] Louis Esperet, Frantiˇsek Kardoˇs, Andrew D. King, Daniel Kr´al, and SergueiNorine. Exponentially many perfect matchings in cubic graphs. Adv. Math. ,227(4):1646–1664, 2011.[7] G. Galbiati. An exact upper bound to the maximum number of perfectmatchings in cubic pseudographs. Calcolo , 18(4):361–370 (1982), 1981.[8] Peter Horak. On graphs with many cycles. Discrete Math. , 331:1–2, 2014.[9] L´aszl´o Lov´asz and Michael D. Plummer.