Mathematics – Combinatorics
Scientific paper
2010-09-04
Mathematics
Combinatorics
10 pages
Scientific paper
Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result vis-\'a-vis the covering radius of sets of permutations, which was studied by Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}). We also conclude with a conjecture of a more general problem in hypergraph matchings.
Aw Alan J.
Ku Cheng Yeaw
No associations
LandOfFree
The covering radius problem for sets of perfect matchings does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The covering radius problem for sets of perfect matchings, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The covering radius problem for sets of perfect matchings will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-284592