Monotonically Decreasing the Number of Directed 3-Cycles via Edge-Flips?
Abstract
We investigate a combinatorial reconfiguration problem on oriented graphs, where a reconfiguration step (edge-flip) is the inversion of the orientation of a single edge. A recently published conjecture that is relevant to the correctness of a Markov Chain Monte Carlo sampler for directed flag complexes states that any simple oriented graph admits a flip sequence that monotonically decreases the number of directed 3-cycles to zero, and is known to be true for complete oriented graphs. We show that, in general, this conjecture does not hold. As main tool for disproving the conjecture, we introduce the concept of FBD-graphs (fully blocked digraphs). An FBD-graph is a directed graph that does not contain any directed 1-, 2-, or 3-cycles, and for which any edge-flip creates a directed 3-cycle. We prove that the non-existence of FBD-graphs is a necessary condition for the conjecture to hold and succeed in constructing FBD-graphs. On the other hand, we show that complete graphs, as well as graphs in which every edge is incident to at most two triangles, cannot be fully blocked. In addition to being relevant for determining in which cases the above mentioned sampling process is correct, the concept of FBD-graphs might also be useful for other problems and yields interesting questions for further study.