101 research outputs found
Group Marriage Problem
Let be a permutation group acting on and
be a system of subsets of . When
is there an element so that for each ? If
such exists, we say that has a -marriage subject to .
An obvious necessary condition is the {\it orbit condition}: for any , for some . Keevash (J. Combin. Theory Ser. A 111(2005),
289--309) observed that the orbit condition is sufficient when is the
symmetric group \Sym([n]); this is in fact equivalent to the celebrated
Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and
only if is a direct product of symmetric groups. We extend the notion of
orbit condition to that of -orbit condition and prove that if is the
alternating group \Alt([n]) or the cyclic group where , then
satisfies the -orbit condition subject to \V if and only if
has a -marriage subject to
Gallai-Edmonds Structure Theorem for Weighted Matching Polynomial
In this paper, we prove the Gallai-Edmonds structure theorem for the most
general matching polynomial. Our result implies the Parter-Wiener theorem and
its recent generalization about the existence of principal submatrices of a
Hermitian matrix whose graph is a tree. keywords:Comment: 34 pages, 5 figure
Cyclic decomposition of k-permutations and eigenvalues of the arrangement graphs
The (n,k)-arrangement graph A(n,k) is a graph with all the k-permutations of
an n-element set as vertices where two k-permutations are adjacent if they
agree in exactly k-1 positions. We introduce a cyclic decomposition for
k-permutations and show that this gives rise to a very fine equitable partition
of A(n,k). This equitable partition can be employed to compute the complete set
of eigenvalues (of the adjacency matrix) of A(n,k). Consequently, we determine
the eigenvalues of A(n,k) for small values of k. Finally, we show that any
eigenvalue of the Johnson graph J(n,k) is an eigenvalue of A(n,k) and that -k
is the smallest eigenvalue of A(n,k) with multiplicity O(n^k) for fixed k.Comment: 18 pages. Revised version according to a referee suggestion
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