The Counting of Perfect Matchings on Some Special Graphs

思齐 杨

doi:10.12677/pm.2022.121004

On Directed Versions of the Hajnal–Szemerédi Theorem

Combinatorics Probability Computing ◽

10.1017/s0963548315000036 ◽

2015 ◽

Vol 24 (6) ◽

pp. 873-928 ◽

Cited By ~ 6

Author(s):

ANDREW TREGLOWN

Keyword(s):

Directed Graph ◽

Minimum Degree ◽

Directed Graphs ◽

Perfect Matchings ◽

Large Order ◽

Vertex Disjoint ◽

Removal Lemma

We say that a (di)graph G has a perfect H-packing if there exists a set of vertex-disjoint copies of H which cover all the vertices in G. The seminal Hajnal–Szemerédi theorem characterizes the minimum degree that ensures a graph G contains a perfect Kr-packing. In this paper we prove the following analogue for directed graphs: Suppose that T is a tournament on r vertices and G is a digraph of sufficiently large order n where r divides n. If G has minimum in- and outdegree at least (1−1/r)n then G contains a perfect T-packing.In the case when T is a cyclic triangle, this result verifies a recent conjecture of Czygrinow, Kierstead and Molla [4] (for large digraphs). Furthermore, in the case when T is transitive we conjecture that it suffices for every vertex in G to have sufficiently large indegree or outdegree. We prove this conjecture for transitive triangles and asymptotically for all r ⩾ 3. Our approach makes use of a result of Keevash and Mycroft [10] concerning almost perfect matchings in hypergraphs as well as the Directed Graph Removal Lemma [1, 6].

Download Full-text

N-processors graphs distributively achieve perfect matchings in O(log2N) beats

Proceedings of the first ACM SIGACT-SIGOPS symposium on Principles of distributed computing - PODC '82 ◽

10.1145/800220.806702 ◽

1982 ◽

Cited By ~ 4

Author(s):

Eli Shamir ◽

Eli Upfal

Keyword(s):

Perfect Matchings

Download Full-text

On the sets of perfect matchings for two bipartite graphs

Information Processing Letters ◽

10.1016/s0020-0190(99)00046-0 ◽

1999 ◽

Vol 70 (2) ◽

pp. 95-97 ◽

Cited By ~ 1

Author(s):

Arnold Rosenbloom

Keyword(s):

Bipartite Graphs ◽

Perfect Matchings

Download Full-text

On the Number of Perfect Matchings in Random Lifts

Combinatorics Probability Computing ◽

10.1017/s0963548309990654 ◽

2010 ◽

Vol 19 (5-6) ◽

pp. 791-817 ◽

Cited By ~ 8

Author(s):

CATHERINE GREENHILL ◽

SVANTE JANSON ◽

ANDRZEJ RUCIŃSKI

Keyword(s):

Asymptotic Formula ◽

Complete Graph ◽

Pairwise Disjoint ◽

Perfect Matching ◽

Perfect Matchings ◽

Lattice Points ◽

Laplace’S Method ◽

Second Moment ◽

Multiple Dimensions ◽

Laplace's Method

Let G be a fixed connected multigraph with no loops. A random n-lift of G is obtained by replacing each vertex of G by a set of n vertices (where these sets are pairwise disjoint) and replacing each edge by a randomly chosen perfect matching between the n-sets corresponding to the endpoints of the edge. Let XG be the number of perfect matchings in a random lift of G. We study the distribution of XG in the limit as n tends to infinity, using the small subgraph conditioning method.We present several results including an asymptotic formula for the expectation of XG when G is d-regular, d ≥ 3. The interaction of perfect matchings with short cycles in random lifts of regular multigraphs is also analysed. Partial calculations are performed for the second moment of XG, with full details given for two example multigraphs, including the complete graph K4.To assist in our calculations we provide a theorem for estimating a summation over multiple dimensions using Laplace's method. This result is phrased as a summation over lattice points, and may prove useful in future applications.

Download Full-text