There are many long-standing open problems on cubic bridgeless graphs, for instance, Jaeger’s directed cycle double cover conjecture. On the other hand, many structural properties of braces have been recently discovered. In this work, we bijectively map the cubic bridgeless graphs to braces which we call the hexagon graphs, and explore the structure of hexagon graphs. We show that hexagon graph...

Let D be a digraph, and let L(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(L(D))= k(D)=0 if every connected component of D is a directed cycle. (2) k(L(D))= k(D)+ n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that ...

We consider the problem of computing a minimum cycle basis in a directed graph G with m arcs and n vertices. The arcs of G have non-negative weights assigned to them. We give an Õ(mn) algorithm, which is the first polynomial time algorithm for this problem. We also present an Õ(mn) randomized algorithm. The problem of computing a minimum cycle basis in an undirected graph has been well-studied....

We consider two variational inequality formulations of the static traac equilibrium problem with xed demand. We determine when an arc-ow formulation can be considered to be equivalent to a path-ow formulation by analyzing the eeect of the form of the travel-time cost function on whether a ow between an origin-destination pair can use all arcs of a directed cycle. An important tool in the analys...