On some Graphs with a Unique Perfect Matching

We show that deciding whether a given graph $G$ of size $m$ has a unique perfect matching as well as finding that matching, if it exists, can be done in time $O(m)$ if $G$ is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching

Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemeredi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs. In particular, we determine the minimum codegree thresholds for Hamilton l-cycles in large k-uniform hypergraphs for l less than k/2. We also determine the minimum vertex degree threshold for loose Hamilton cycle in large 3-uniform hypergraphs. These results generalize the well-known theorem of Dirac for graphs. Third, we determine the minimum codegree threshold for near perfect matchings in large k-uniform hypergraphs, thereby confirming a conjecture of Rodl, Rucinski and Szemeredi. We also show that the decision problem on whether a k-uniform hypergraph with certain minimum codegree condition contains a perfect matching can be solved in polynomial time, which solves a problem of Karpinski, Rucinski and Szymanska completely. At last, we determine the minimum vertex degree threshold for perfect tilings of C_4^3 in large 3-uniform hypergraphs, where C_4^3 is the unique 3-uniform hypergraph on four vertices with two edges

Quantum Algorithms for Memoryless Search and Perfect Matching

In this thesis, we present two new quantum algorithms for graph problems. The first algorithm we give is a memoryless walk that can find a unique marked vertex on a two-dimensional grid. Our walk is based on a construction proposed by Falk, which tessellates the grid with squares of size 2 × 2. Our walk uses minimal memory, O(sqrt(N logN)) applications of the walk operator, and outputs the marked vertex with vanishing error probability. To accomplish this, we apply a selfloop to the marked vertex—a technique we adapt from interpolated walks. We prove that with our explicit choice of selfloop weight, this forces the action of the walk asymptotically into a single rotational space. We characterize this space and as a result, show that our memoryless walk produces the marked vertex with a success probability asymptotically approaching one. Our second algorithm decides whether a graph contains a perfect matching. This is the first quantum algorithm based on the algebraic characterization by Tutte, which reduces the problem of detecting perfect matchings to deciding whether a matrix has nonzero determinant. The key part of our algorithm is a new span program that can decide whether a matrix is singular. Our span program has a simple structure and its witness size matches that of a related span program by Belovs for matrix rank-finding, up to a constant factor. Using a transformation given by Reichardt, our span program can be compiled into a quantum algorithm, which we use as a subroutine in our algorithm to detect perfect matchings. We also show that there are families of graphs for which our perfect matching detection algorithm may have exponential query complexity. These graphs could be a useful tool in determining the tight quantum query complexity of the perfect matching detection problem, which remains an open problem

Computing Unique Maximum Matchings in O(m) time for Konig-Egervary Graphs and Unicyclic Graphs

Let alpha(G) denote the maximum size of an independent set of vertices and mu(G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is perfect. If alpha(G) + mu(G) equals the number of vertices of G, then it is called a Konig-Egervary graph. A graph is unicyclic if it has a unique cycle. In 2010, Bartha conjectured that a unique perfect matching, if it exists, can be found in O(m) time, where m is the number of edges. In this paper we validate this conjecture for Konig-Egervary graphs and unicylic graphs. We propose a variation of Karp-Sipser leaf-removal algorithm (Karp and Spiser, 1981), which ends with an empty graph if and only if the original graph is a Konig-Egervary graph with a unique perfect matching obtained as an output as well. We also show that a unicyclic non-bipartite graph G may have at most one perfect matching, and this is the case where G is a Konig-Egervary graph.Comment: 10 pages, 5 figure

On the expected number of perfect matchings in cubic planar graphs

A well-known conjecture by Lov\'asz and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. (Adv. Math. 2011). On the other hand, Chudnovsky and Seymour (Combinatorica 2012) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c\gamma^n$, where $c>0$ and $\gamma \sim 1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (non necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.Comment: 19 pages, 4 figure