How many subsets of edges of a directed multigraph can be represented as trails?

TL;DR

This paper investigates the proportion of edge subsets in directed multigraphs that can be represented as trails, establishing an upper bound that approaches tightness as the number of edges grows.

Contribution

It provides a theoretical upper bound on the fraction of trail-representable edge subsets in directed multigraphs, with near-tightness demonstrated through examples.

Findings

The fraction of trail-representable subsets is at most O(√(log m)/√m).

The upper bound is nearly tight, as shown by specific examples.

The result advances understanding of trail representations in complex graphs.

Abstract

For each subset of edges of a (directed multi-) graph, one may determine whether the edges can be represented as a trail. I prove that the fraction trail-representable subsets of edges is at most $O(\sqrt{\log m}/\sqrt{m})$, where $m$ is the number of edges, and show by example that the upper bound is nearly tight.