Perfect Matchings in ε-regular Graphs

A super (d, )-regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2, where |V1| = |V2| = n, in which the minimum degree and the maximum degree are between (d− )n and (d+ )n, and for every U ⊂ V1,W ⊂ V2 with |U | ≥ n, |W | ≥ n, | e(U,W ) |U ||W | − e(V1,V2) |V1||V2| | < . We prove that for every 1 > d > 2 > 0 and n > n0( ), the number of perfect matchings in any such graph is at least (d − 2 )nn! and at most (d + 2 )nn!. The proof relies on the validity of two well known conjectures for permanents; the Minc conjecture, proved by Brégman, and the van der Waerden conjecture, proved by Falikman and Egorichev. An -regular graph on 2n vertices is a bipartite graph on the classes of vertices V1 and V2, where |V1| = |V2| = n, in which for every U ⊂ V1,W ⊂ V2 with |U | ≥ n, |W | ≥ n, ∣∣∣∣e(U,W ) |U ||W | − e(V1, V2) |V1||V2| ∣∣∣∣ < , (1) ∗Research supported in part by a USA Israeli BSF grant, by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University and by a State of New Jersey grant. †Research supported by Polish-US NSF grant INT-940671 and by NSF grant DMS-9704114. ‡Research supported by Polish-US NSF grant INT-940671 and by KBN grant 2 P03A 023 09. Mathematics Subject Classification (1991); primary 05C50, 05C70; secondary 05C80