Decomposing a signed graph into rooted circuits

TL;DR

This paper establishes a min-max theorem for decomposing Eulerian graphs into rooted circuits with specific edge conditions, connecting to vertex-minors and addressing conjectures in graph theory.

Contribution

It provides a precise min-max characterization for partitioning Eulerian graphs into rooted trails with odd-signed edges, linking to vertex-minor problems and conjectures.

Findings

Proves a min-max theorem for rooted circuit decompositions.

Connects the decomposition problem to vertex-minors.

Derives two conjectures of Mácajová and Škoviera as corollaries.

Abstract

We prove a precise min-max theorem for the following problem. Let $G$ be an Eulerian graph with a specified set of edges $S \subseteq E(G)$, and let $b$ be a vertex of $G$. Then what is the maximum integer $k$ so that the edge-set of $G$ can be partitioned into $k$ non-zero $b$-trails? That is, each trail must begin and end at $b$ and contain an odd number of edges from~$S$. This theorem is motivated by a connection to vertex-minors and yields two conjectures of M\'{a}\v{c}ajov\'{a} and \v{S}koviera as corollaries.