Some conjectures on \(r\)-graphs and equivalences

An \(r\)-regular graph is an \(r\)-graph, if every odd set of vertices is connected to its complement by at least \(r\) edges. Seymour [On multicolourings of cubic graphs, and conjectures of Fulkerson and Tutte.~\emph{Proc.~London Math.~Soc.}~(3), 38(3): 423-460, 1979] conjectured (1) that every planar \(r\)-graph is \(r\)-edge colorable and (2) that every \(r\)-graph has \(2r\) perfect matchings such that every edge is contained in precisely two of them. We study several variants of these conjectures. A \((t,r)\)-PM is a multiset of \(t \cdot r\) perfect matchings of an \(r\)-graph \(G\) such that every edge is in precisely \(t\) of them. We show that the following statements are equivalent for every \(t, r \geq 1\): 1. Every planar \(r\)-graph has a \((t,r)\)-PM. 2. Every \(K_5\)-minor-free \(r\)-graph has a \((t,r)\)-PM. 3. Every \(K_{3,3}\)-minor-free \(r\)-graph has a \((t,r)\)-PM. 4. Every \(r\)-graph whose underlying simple graph has crossing number at most \(1\) has a \((t,r)\)-PM.