Perfect matchings in highly cyclically connected regular graphs

LetG=G(n) be a randomly chosenk-edge-colouredk-regular graph with 2nvertices, wherek=k(n). Such a graph can be obtained from a random set ofkedge-disjoint perfect matchings ofK2n. Leth=h(n) be a graph withm=m(n) edges such thatm2+mk=o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs ofGisomorphic toh. Isomorphisms may or may not respect the edge colouring, and other generalizations are also presented. Special attention is paid to matchings and cycles.The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number ofk-edge-colouredk-regular graphs fork=o(n5/6) is found.

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The number of disjoint perfect matchings in semi-regular graphs

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm161109030l ◽

2017 ◽

Vol 11 (1) ◽

pp. 11-38

Author(s):

Hongliang Lu ◽

David Wang

Keyword(s):

Regular Graph ◽

Perfect Matchings ◽

Regular Graphs ◽

Sharp Result

We obtain a sharp result that for any even n ? 34, every {Dn,Dn+1}-regular graph of order n contains ?n/4? disjoint perfect matchings, where Dn = 2?n/4?-1. As a consequence, for any integer D ? Dn, every {D, D+1}- regular graph of order n contains (D-?n/4?+1) disjoint perfect matchings.

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Perfect Matchings in ϵ-Regular Graphs and the Blow-Up Lemma

COMBINATORICA ◽

10.1007/s004930050063 ◽

1999 ◽

Vol 19 (3) ◽

pp. 437-452 ◽

Cited By ~ 31

Author(s):

Vojtech Rödl ◽

Andrzej Ruciński

Keyword(s):

Blow Up ◽

Perfect Matchings ◽

Regular Graphs

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Perfect Matchings in $\epsilon$-regular Graphs

The Electronic Journal of Combinatorics ◽

10.37236/1351 ◽

1998 ◽

Vol 5 (1) ◽

Cited By ~ 4

Author(s):

Noga Alon ◽

Vojtech Rödl ◽

Andrzej Ruciński

Keyword(s):

Bipartite Graph ◽

Regular Graph ◽

Maximum Degree ◽

Minimum Degree ◽

Perfect Matchings ◽

Regular Graphs

A super $(d,\epsilon)$-regular graph on $2n$ vertices is a bipartite graph on the classes of vertices $V_1$ and $V_2$, where $|V_1|=|V_2|=n$, in which the minimum degree and the maximum degree are between $ (d-\epsilon)n$ and $ (d+\epsilon) n$, and for every $U \subset V_1, W \subset V_2$ with $|U| \geq \epsilon n$, $|W| \geq \epsilon n$, $|{{e(U,W) }\over{|U||W|}}-{{e(V_1,V_2)}\over{|V_1||V_2|}}| < \epsilon.$ We prove that for every $1>d >2 \epsilon >0$ and $n>n_0(\epsilon)$, the number of perfect matchings in any such graph is at least $(d-2\epsilon)^n n!$ and at most $(d+2 \epsilon)^n n!$. 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.

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Scheduling Partial Round Robin Tournaments Subject to Home Away Pattern Sets

The Electronic Journal of Combinatorics ◽

10.37236/144 ◽

2009 ◽

Vol 16 (1) ◽

Cited By ~ 1

Author(s):

Kenji Kashiwabara

Keyword(s):

Regular Graph ◽

Perfect Matchings ◽

Integral Solution ◽

Linear Inequalities ◽

Necessary Condition ◽

Scheduling Problem ◽

Regular Graphs ◽

Round Robin Tournaments ◽

Sport League ◽

Home Game

We consider the following sports scheduling problem. Consider $2n$ teams in a sport league. Each pair of teams must play exactly one match in $2n-1$ days. That is, $n$ games are held simultaneously in a day. We want to make a schedule which has $n(2n-1)$ games for $2n-1$ days. When we make a schedule, the schedule must satisfy a constraint according to the HAP set, which designates a home game or an away game for each team and each date. Two teams cannot play against each other unless one team is assigned to a home game and the other team is assigned to an away game. Recently, D. Briskorn proposed a necessary condition for an HAP set to have a proper schedule. And he proposed a conjecture that such a condition is also sufficient. That is, if a solution to the linear inequalities exists, they must have an integral solution. In this paper, we rewrite his conjecture by using perfect matchings. We consider a monoid in the affine space generated by perfect matchings. In terms of the Hilbert basis of such a monoid, the problem is naturally generalized to a scheduling problem for not all pairs of teams described by a regular graph. In this paper, we show a regular graph such that the corresponding linear inequalities have a solution but do not have any integral solution. Moreover we discuss for which regular graphs the statement generalizing the conjecture holds.

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