Circuits through prescribed edges

TL;DR

This paper characterizes when a connected graph contains a circuit passing through any specified set of edges, linking this to the absence of small odd cuts, thus providing a precise combinatorial condition.

Contribution

It establishes a necessary and sufficient condition for the existence of circuits through prescribed edges based on odd cut sizes.

Findings

A connected graph contains a circuit through any $k$ prescribed edges iff it has no odd cut of size at most $k$.

The result provides a new combinatorial characterization of circuits through prescribed edges.

The theorem generalizes previous results on circuits and cuts in graphs.

Abstract

We prove that a connected graph contains a circuit---a closed walk that repeats no edges---through any $k$ prescribed edges if and only if it contains no odd cut of size at most $k$.