Monotonically Decreasing the Number of Directed 3-Cycles via Edge-Flips?

TL;DR

This paper investigates a conjecture about reducing directed 3-cycles in graphs through edge-flips, disproves it in general, introduces FBD-graphs as a key concept, and explores conditions for the conjecture's validity.

Contribution

The paper introduces FBD-graphs to analyze the conjecture, disproves the conjecture in general, and identifies conditions under which the conjecture may hold.

Findings

Disproved the conjecture for general graphs.

Introduced FBD-graphs as a tool for analysis.

Identified classes of graphs that cannot be fully blocked.

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.