A directed graph has a natural Z-module homomorphism from the underlying graph’s cycle space to Z where the image of an oriented cycle is the number of forward edges minus the number of backward edges. Such a homomorphism preserves the parity of the length of a cycle and the image of a cycle is bounded by the length of that cycle. Pretzel and Youngs [1] showed that any Z-module homomorphism of ...

The problem of finding a negative cycle in a weighted, directed graph is discussed here. First the algorithm for printing out a negative cycle reachable from the source s, with the running time no worse than the Bellman-Ford algorithm for the single-source shortest-path problem, is presented. Then an approach with the same time complexity, which could be used for outputting a negative cycle tha...

Let G = Gn(a1, a2, ..., ak) denote a directed circulant graph of order n with k pairwise distinct jumps [1]. Antidirected Hamilton cycle in G is a cycle of n arcs that does not contain induced directed path P2. Let C 1 G, C 2 G, ..., C k G be pairwise arc-disjoint antidirected Hamilton cycles in G, each composed of exactly two distinct jumps. We give the necessary and sufficient conditions for ...

Recent progress in our understanding of the human immunodeficiency virus type 1 (HIV-1) life cycle has lead to the identification and characterization of viral genes or gene products that have been evaluated as targets for gene therapy. Virtually every stage in the viral life cycle and every viral gene product is a potential target. Gene therapy approaches directed at several of these viral tar...

A cycle cover of a graph is a spanning subgraph where each node is part of exactly one simple cycle. A k-cycle cover is a cycle cover where each cycle has length at least k. We call the decision problems whether a directed or undirected graph has a k-cycle cover k-DCC and k-UCC. Given a graph with edge weights one and two, Min-k-DCC and Min-k-UCC are the minimization problems of finding a k-cyc...

An orientation of a graph G is Pfaffian if every even cycle C such that G\V (C) has a perfect matching has an odd number of edges directed in either direction of the cycle. The significance of Pfaffian orientations is that if a graph has one, then the number of perfect matchings (a.k.a. the dimer problem) can be computed in polynomial time. The question of which bipartite graphs have Pfaffian o...

Phylogenetic networks are becoming increasingly popular in phylogenetics since they have the ability to describe a wider range of evolutionary events than their tree counterparts. In this paper, we study Markov models on phylogenetic networks and their associated geometry. We restrict our attention to large-cycle networks, networks with a single undirected cycle of length at least four. Using t...

A constant-round interactive argument is introduced to show existence of a Hamiltonian cycle in a directed graph. Graph is represented with a characteristic polynomial, top coefficient of a verification polynomial is tested to fit the cycle, soundness follows from Schwartz-Zippel lemma.

We show that every digraph D with minimum outdegree 6 and maximum indegree A contains a directed cycle of length O(mod k), provided e[dd+ l](1-l/k) " < 1. In particular, if A < (2 "-e)/e6 then D contains an even cycle. 0 1989 Academic Press, Inc.