Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant $0<\lambda<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p^{\lfloor (1-\lambda)\frac{n}{3p}\rfloor}$ in expected time less than $c^n$ for a constant $c<2$ that depends only on $p$ and $\lambda$. Such an algorithm was previously known only for the case of counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in $O^*(3^{n-\alpha(G)})$ time and polynomial space, where $\alpha(G)$ is the size of the maximum independent set in $G$. In particular, this yields an $O^*(3^{n/2})$ time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an $O^*(2^k)$-time randomized algorithm for detecting out-branchings with at least $k$ internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed $k$-Leaf problem, based on a non-standard monomial detection problem

On non-Hamiltonian circulant digraphs of outdegree three

We construct infinitely many connected, circulant digraphs of outdegree three that have no hamiltonian circuit. All of our examples have an even number of vertices, and our examples are of two types: either every vertex in the digraph is adjacent to two diametrically opposite vertices, or every vertex is adjacent to the vertex diametrically opposite to itself

On Cayley digraphs that do not have hamiltonian paths

We construct an infinite family of connected, 2-generated Cayley digraphs Cay(G;a,b) that do not have hamiltonian paths, such that the orders of the generators a and b are arbitrarily large. We also prove that if G is any finite group with |[G,G]| < 4, then every connected Cayley digraph on G has a hamiltonian path (but the conclusion does not always hold when |[G,G]| = 4 or 5).Comment: 10 pages, plus 14-page appendix of notes to aid the refere