aa r X i v : . [ m a t h . C O ] J un Covering a cubic graph with perfect matchings
G.Mazzuoccolo ∗ September 10, 2018
Abstract
Let G be a bridgeless cubic graph. A well-known conjecture of Berge andFulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, weprove that in each bridgeless cubic graph there exist five perfect matchingscovering a portion of the edges at least equal to . By a generalization ofthis result, we decrease the best known upper bound, expressed in termsof the size of the graph, for the number of perfect matchings needed tocover the edge-set of G . Keywords: Berge-Fulkerson conjecture, perfect matchings, cubic graphs.MSC(2010): 05C15 (05C70)
Throughout this paper, a graph G always means a simple connected finite graph(without loops and parallel edges). Furthermore, along the entire paper G stands for a bridgeless cubic graph, unless otherwise specified, and we denoteby V ( G ) and E ( G ) the vertex-set and the edge-set of G , respectively. A perfectmatching of G is a 1-regular spanning subgraph of G . Following the definitionintroduced in [8] and [12], we define m t ( G ) to be the maximum fraction of theedges in G that can be covered by t perfect matchings, and by m t the infimumof all m t ( G ) over all bridgeless cubic graphs. That is, m t = inf G max M ,...,M t | S ti =1 M i || E ( G ) | The Berge–Fulkerson conjecture, one of the challenging open problems ingraph theory, can be easily stated in terms of m t : it is the assertion that m = 1 (see [10]).Kaiser, Kr´al and Norine [8] proved that m = and m ≥ . In a fi-nal remark of their paper, they announced, without a proof, the more general ∗ Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Universit`a di Modena eReggio Emilia, via Campi 213/b, 41125 Modena (Italy) a t ≤ m t , where a t is the sequence defined by the recurrence a t = t t + 1 (1 − a t − ) + a t − (1)and a = 0.In the present paper we present a complete proof of the result announcedin [8], and we use it to deduce a new upper bound in terms of t for the size ofa bridgeless cubic graph admitting a covering with t perfect matchings. Moreprecisely, it was known (see for instance [12]) that a bridgeless cubic graph withfewer than ( ) t edges can be covered using t perfect matchings. We improve thisbound by proving that each bridgeless cubic graph with fewer than t √ t edgescan be covered with t perfect matchings.Finally, as a by-product of our main result, we also obtain that in eachbridgeless cubic graph there is a set of t perfect matchings with no (2 t + 1)-cut in their intersection, giving partial support to a conjecture of Kaiser andRaspaud [7], in a special form due to M´acajov´a and Skoviera [9], about theexistence of two perfect matchings with no odd cut in their intersection.The main tool for our proof is the Perfect Matching Polytope Theorem ofEdmonds (see [2]); we briefly recall it in the next section. Let G be a graph. A minimal cut C in G is a subset of E ( G ) such that G \ C has more components than G does, and C is inclusion-wise minimal with thisproperty. A k -cut is a minimal cut of cardinality k . When X ⊆ V ( G ), let ∂X denote the set of edges with precisely one end in X .Let w be a vector in R E ( G ) . The entry of w corresponding to an edge e is denoted by w ( e ), and for A ⊆ E ( G ), we define the weight w ( A ) of A as P e ∈ A w ( e ). The vector w is a fractional perfect matching of G if it satisfies thefollowing properties:a) 0 ≤ w ( e ) ≤ e ∈ E ( G ),b) w ( ∂ { v } ) = 1 for each vertex v ∈ V , andc) w ( ∂X ) ≥ X ⊆ V ( G ) of odd cardinality.We will denote by P ( G ) the set of all fractional perfect matchings of G . If M is a perfect matching, then the characteristic vector χ M ∈ R E of M is containedin P ( G ). Furthermore, if w , . . . , w n ∈ P ( G ), then any convex combination P ni =1 α i w i also belongs to P ( G ). It follows that P ( G ) contains the convex hullof all vectors χ M such that M is a perfect matching of G . The Perfect MatchingPolytope Theorem of Edmonds asserts that the converse inclusion also holds: Theorem 2.1 (Edmonds) . For any graph G , the set P ( G ) is precisely theconvex hull of the characteristic vectors of perfect matchings of G . Lemma 2.1. If w is a fractional perfect matching in a graph G and c ∈ R E ,then G has a perfect matching M such that c · χ M ≥ c · w where · denotes the dot product. Moreover, there exists such a perfect matching M that contains exactly one edge of each odd cut C with w ( C ) = 1 . A proof of this lemma can be found in [8]. m t In this section we prove m t ≥ a t for each index t , where a t is the sequencedefined by the recurrence a t = t t + 1 (1 − a t − ) + a t − (2)and a = 0. In particular, we deduce m ≥ .Let M t = { M , . . . , M t } , where each M i is a perfect matching of a bridgelesscubic graph G , and set M = ∅ .For each subset A of the edge-set of G we defineΦ( A, M t ) = t X i =1 | A ∩ M i | . Furthermore, define the weight w M t by w M t ( e ) = t + 1 − P ti =1 | M i ∩ { e }| t + 3 = t + 1 − Φ( { e } , M t )2 t + 3 . When |M| = 1 or 2, we obtain the spaecial cases w and w in the maintheorem of [8].Hence, for a set A ⊆ E ( G ), with | A | = k , the weight of A is given by thefollowing relation: w M t ( A ) = k ( t + 1) − Φ( A, M t )2 t + 3 . In other words, by the definition of the weight w M t , the weight of a setdepends only on the size of the intersections with the perfect matchings in M t .An easy calculation proves the following lemma,3 emma 3.1. For A ⊆ E ( G ) with | A | = k . w M t ( A ) ≥ ⇐⇒ Φ( A, M t ) ≤ t ( k −
2) + ( k − and equality holds on one side of the implication if and only if it holds on theother side. We are now ready to state the main result.
Theorem 3.1.
Let a t be the sequence (2). Then, m t ≥ a t for each index t .Proof. We argue by induction: let M t = { M , . . . , M t } be a set of t perfectmatchings such that w M t is a fractional perfect matching of a bridgeless cubicgraph G . Now, we construct a set M t +1 of t + 1 perfect matchings and suchthat w M t +1 is a fractional perfect matching of G . Set c t = 1 − χ S ti =1 M i . ByLemma 2.1 there exists a perfect matching M t +1 such that c t · χ M t +1 ≥ c t · w M t and such that M t +1 contains exactly one edge for each cut C with w M t ( C ) = 1.In order to prove that w M t +1 , where M t +1 = M t ∪ { M t +1 } , is a fractionalperfect matching of G , we have to verify the properties ( a ) , ( b ) and ( c ) of thedefinition of fractional perfect matching.( a ) 0 ≤ w M t +1 ( e ) ≤ e ∈ E ( G ).( b ) Let v be a vertex of G . We have w M t +1 ( ∂ { v } ) = 3( t + 2) − Φ( ∂ { v } , M t +1 )2 t + 5since each perfect matching M i intersects ∂ { v } exactly one time we have Φ( ∂ { v } , M t +1 ) = t + 1 and then w M t +1 ( ∂ { v } ) = 3( t + 2) − ( t + 1)2 t + 5 = 1 . ( c ) Let C be a k -cut of G , with k odd. w M t +1 ( C ) = k ( t + 2) − Φ( C, M t +1 )2 t + 5If k = 3, then by the assumption that w M t is a fractional perfect matching,we have w M t ( C ) = 3( t + 1) − Φ( C, M t )2 t + 3 ≥ C, M t ) ≤ t = |M t | . Furthermore, a 3-cut intersects each of the t per-fect matchings M i , hence Φ( C, M t ) ≥ t . Then Φ( C, M t ) = t and w M t ( C ) = 1.By Lemma 2.1, we obtain | C ∩ M t +1 | = 1. Now, we can compute Φ( C, M t +1 ) =Φ( C, M t ) + | C ∩ M t +1 | = t + 1. We have proved that w M t +1 ( C ) = 3( t + 2) − Φ( C, M t +1 )2 t + 5 = 3( t + 2) − ( t + 1)2 t + 5 = 1 ,
4s required.If k >
3, we distinguish two cases according that w M t ( C ) = 1 (that isΦ( C, M t ) = t ( k −
2) + ( k − w M t ( C ) > C, M t ) < t ( k −
2) +( k − | C ∩ M t +1 | = 1, soΦ( C, M t +1 ) = Φ( C, M t )+ | C ∩ M t +1 | = t ( k − k − t +1)( k − ≤ ( t +1)( k − k − . Now w M t +1 ( C ) ≥ | M t +1 ∩ C | ≤ k , henceΦ( C, M t +1 ) = Φ( C, M t ) + | C ∩ M t +1 | < t ( k −
2) + ( k −
3) + k. Since each perfect matching intersects an odd cut an odd number of timesfollows Φ( C, M t ) ≡ t (mod 2), so the previous inequality is equivalent toΦ( C, M t +1 ) ≤ t ( k −
2) + ( k −
5) + k = ( t + 1)( k −
2) + ( k − . By Lemma 3.1, we can infer w M t +1 ( C ) ≥ w M ( e ) = is trivially a fractional perfectmatching, we have that w M t (with M t constructed as described above) is afractional perfect matching for each value of t . Therefore, the following holds: c t · χ M t +1 ≥ c t · w M t . The left side of the previous inequality is exactly the number of edges of M t +1 not covered by M t , while the right side is t t +1 times the number of edges notcovered by M t . Denoting by a t the fraction of edges of E ( G ) covered by M t ,we obtain m t +1 − a t ≥ a t +1 − a t = (1 − a t ) · t + 12 t + 3 ,m t +1 ≥ (1 − a t ) · t + 12 t + 3 + a t = a t +1 and the assertion follows.By direct calculation, a = , and the following corollary holds: Corollary 3.1.
Let G be a bridgeless cubic graph. There exist five perfectmatchings of G that cover at least ⌈ | E ( G ) |⌉ edges of G . Furthermore, we obtain the following corollary by the proof of Theorem 3.1.
Corollary 3.2.
Let G be a bridgeless cubic graph and t a positive integer. Thereexist t perfect matchings of G with no (2 t + 1) -cut in their intersection. A bound for the size of a bridgeless cubicgraph which admits a covering with t perfectmatchings As remarked in [12], it is unknown whether m t = 1 for any t ≥
5. The bestknown result in this direction is the following: if G is a bridgeless cubic graphof size | E ( G ) | , then m t = 1 when t > log ( E ). In other words the edge-setof each bridgeless cubic graph with fewer than ( ) t edges can be covered by t perfect matchings. The following improvement of that bound is a consequenceof Theorem 3.1. Theorem 4.1. If G is a bridgeless cubic graph with fewer than t √ t edges, thenthere is a covering of G by t perfect matchings.Proof. Let G be a bridgeless cubic graph of size | E ( G ) | . It is trivial that if | E ( G ) | m t > | E ( G ) | −
1, that is | E ( G ) | < − m t , then there exists a covering of G by t perfect matchings. The inequality − a t ≤ − m t follows by Theorem 3.1.Therefore the theorem is proved, if we prove that the inequality ⌊ t √ t ⌋ < − a t holds for each t >
0. One can easily verify that the inequality holds for t < t √ t < − a t holds for each t ≥ t = 5, we have by direct computation32 √ < − a = 23116 . Suppose t √ t < − a t . We have a t +1 = t + 12 t + 3 (1 − a t ) + a t > t + 12 t + 3 + t + 22 t + 3 (1 − √ t t ) = 1 − t + 22 t + 3 √ t t ;the assertion follows by a direct check that t + 22 t + 3 √ t t < √ t + 12 t +1 holds for each t . We would like to stress that the special cases m ≥ a = and m ≥ a = of Theorem 3.1 were already proved in [8]. Furthermore, it can be immediatelychecked that the Petersen graph realizes the equality m = a . Mainly for thisreason, it is conjectured in [12] that also m and m reach their minimum when G is the Petersen graph.It is trivial that every possible counterexample to the Berge–Fulkerson con-jecture must be a snark, if it exists: a stronger version of the Berge–Fulkerson6onjecture could be that m ( G ) = 1 for each snark other than the Petersengraph (see [4]). Essentially the Petersen graph should be the unique obstruc-tion to obtaining a covering with at most four perfect matchings . Followingthis point of view, Bonvicini and the author prove in [1] that a large class ofbridgeless cubic graphs (including the Petersen graph) of order 2 n can be cov-ered by at most 4 matchings of size n − r -regular graph defined in [13]: Seymour proposed a naturalgeneralization of the Berge-Fulkerson conjecture for this class of graphs and itturns out that his conjecture is equivalent to the assertation that each graph G of this class can be covered with at most 2 r − r -regular graph, for r >
3, can be covered with t perfect matchings. The author is grateful to the reviewers and the editor for their very usefulcomments and advices.