A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, t
hen $|mathcal{F}| leq (2(n-1) - 1)!!$ and if equality holds, then $mathcal{F} = mathcal{F}_{ij}$ where $ mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family $mathcal{F}$ is non-Hamiltonian, that is, $m cup m ot cong C_{2n}$ for any $m,m in mathcal{F}$, then $|mathcal{F}| leq (2(n-1) - 1)!!$ and this bound is met with equality if and only if $mathcal{F} = mathcal{F}_{ij}$. Our results make ample use of a somewhat understudied symmetric commutative association scheme arising from the Gelfand pair $(S_{2n},S_2 wr S_n)$. We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.
Given a graph $G$, a vertex switch of $v in V(G)$ results in a new graph where neighbors of $v$ become nonneighbors and vice versa. This operation gives rise to an equivalence relation over the set of labeled digraphs on $n$ vertices. The equivalence
class of $G$ with respect to the switching operation is commonly referred to as $G$s switching class. The algebraic and combinatorial properties of switching classes have been studied in depth; however, they have not been studied as thoroughly from an algorithmic point of view. The intent of this work is to further investigate the algorithmic properties of switching classes. In particular, we show that switching classes can be used to asymptotically speed up several super-linear unweighted graph algorithms. The current techniques for speeding up graph algorithms are all somewhat involved insofar that they employ sophisticated pre-processing, data-structures, or use word tricks on the RAM model to achieve at most a $O(log(n))$ speed up for sufficiently dense graphs. Our methods are much simpler and can result in super-polylogarithmic speedups. In particular, we achieve better bounds for diameter, transitive closure, bipartite maximum matching, and general maximum matching.
Lekkerkerker and Boland characterized the minimal forbidden induced subgraphs for the class of interval graphs. We give a linear-time algorithm to find one in any graph that is not an interval graph. Tucker characterized the minimal forbidden submatr
ices of binary matrices that do not have the consecutive-ones property. We give a linear-time algorithm to find one in any binary matrix that does not have the consecutive-ones property.
A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. A partially complemented digraph $widetilde{G}$ is a digraph obtained from a sequence of vertex complement operations on $G
$. Dahlhaus et al. showed that, given an adjacency-list representation of $widetilde{G}$, depth-first search (DFS) on $G$ can be performed in $O(n + widetilde{m})$ time, where $n$ is the number of vertices and $widetilde{m}$ is the number of edges in $widetilde{G}$. To achieve this bound, their algorithm makes use of a somewhat complicated stack-like data structure to simulate the recursion stack, instead of implementing it directly as a recursive algorithm. We give a recursive $O(n+widetilde{m})$ algorithm that uses no complicated data-structures.
