Packing odd $T$-joins with at most two terminals

TL;DR

This paper proves a packing conjecture for signed graphs with two terminals, establishing conditions under which maximum edge-disjoint odd T-joins match minimum cut or signature sizes, with implications for various graph classes.

Contribution

It confirms the Cycling Conjecture for odd T-joins with at most two terminals by characterizing packings via forbidden minors in Eulerian signed grafts.

Findings

Confirmed the Cycling Conjecture for this class.

Characterized weakly and evenly bipartite graphs.

Provided new bounds for edge covering with cuts.

Abstract

Take a graph $G$, an edge subset $\Sigma\subseteq E(G)$, and a set of terminals $T\subseteq V(G)$ where $|T|$ is even. The triple $(G,\Sigma,T)$ is called a signed graft. A $T$-join is odd if it contains an odd number of edges from $\Sigma$. Let $\nu$ be the maximum number of edge-disjoint odd $T$-joins. A signature is a set of the form $\Sigma\triangle \delta(U)$ where $U\subseteq V(G)$ and $|U\cap T)$ is even. Let $\tau$ be the minimum cardinality a $T$-cut or a signature can achieve. Then $\nu\leq \tau$ and we say that $(G,\Sigma,T)$ packs if equality holds here. We prove that $(G,\Sigma,T)$ packs if the signed graft is Eulerian and it excludes two special non-packing minors. Our result confirms the Cycling Conjecture for the class of clutters of odd $T$-joins with at most two terminals. Corollaries of this result include, the characterizations of weakly and evenly bipartite graphs, packing two-commodity paths, packing $T$-joins with at most four terminals, and a new result on covering edges with cuts.