On the Number of Matchings in Regular Graphs

Abstract

For the set of graphs with a given degree sequence, consisting of any number of 2 ′ s and 1 ′ s, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of m-matchings.We find the expected value of the number of m-matchings of r-regular bipartite graphs on 2n vertices with respect to the two standard measures. We state and discuss the conjectured upper and lower bounds for m-matchings in r-regular bipartite graphs on 2n vertices, and their asymptotic versions for infinite r-regular bipartite graphs. We prove these conjectures for 2-regular bipartite graphs and for m-matchings with m ≤ 4.2000 Mathematics Subject Classification: 05A15, 05A16, 05C70, 05C80, 82B20 Keywords and phrases: Partial matching and asymptotic growth of average matchings for r-regular bipartite graphs, asymptotic matching conjectures.1 The object of this paper is two folds. First we consider the family Ω(n, k), the set of simple graphs on n vertices with 2k vertices of degree 1 and n − 2k vertices of degree 2. Let Ω bi (n, k) ⊂ Ω(n, k) be the subset of bipartite graphs. For each m ∈ [2, n] ∩ N we characterize the optimal graphs which maximize and minimize φ(m, G), m ≥ 2 for G ∈ Ω(n, k) and G ∈ Ω bi (n, k). It turns out the optimal graphs do not depend on m but on n and k. Furthermore, the graphs with the maximal number of m-matchings, are bipartite.Second, we consider G(2n, r), the set of simple bipartite r-regular graphs on 2n vertices, where n ≥ r. Denote by C l a cycle of length l and by K r,r the complete bipartite graph with r-vertices in each group. For a nonnegative integer q and a graph G denote by qG the disjoint union of q copies of G. Let λ(m, n, r) := min G∈G(2n,r) φ(m, G), Λ(m, n, r) := max G∈G(2n,r) φ(m, G), m = 1, . . . , n. (1.1) Our results on 2-regular graphs yield. λ(m, n, 2) = φ(m, C 2n ), (1.2) Λ(m, 2q, 2) = φ(m, qK 2,2 ), Λ(m, 2q + 3, 2) = φ(m, qK 2,2 ∪ C 6 ), (1.3)for m = 1, . . . , n.The equality Λ(m, 2q, 2) = φ(m, qK 2,2 ) inspired us to conjecture the Upper Matching Conjecture, abbreviated here as UMC: Λ(m, qr, r)) = φ(m, qK r,r ) for m = 1, . . . , qr.( 1.4) For the value m = qr the UMC follows from Bregman's inequality [1]. For the value r = 3 the UMC holds up to q ≤ 8. The results of [4] support the validity of the above conjecture for r = 3, 4 and large values of n. As in the case r = 2 we conjecture that that for any nonbipartite r-regular graph on 2n vertices φ(m, G) ≤ Λ(m, n, r) for m = 1, . . . , n.It is useful to consider G mult (2n, r) ⊃ G(2n, r), the set of r-regular bipartite graphs on 2n vertices, where multiple edges are allowed. Observe that G mult (2, r) = {H r }, where H r is the r-regular multi-bipartite graph on 2 vertices. Let µ(m, n, r) := min G∈G mult (2n,r) φ(m, G), M (m, n, r) := max G∈G mult (2n,r) φ(m, G), (1.5) m = 1, . . . , n, 2 ≤ r ∈ N.It is straightforward to show that M (m, n, r) = φ(m, nH r ) = n m r m , m = 1, . . . , n.(1.6)Hence for most of the values of m Λ(m, n, r) < M (m, n, r). On the other hand, as in the case of Ω(n, k), it i...

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