Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor

Abstract: An $r$-regular graph is an $r$-graph, if every odd set of vertices is connected to its complement by at least $r$ edges. We prove for $r \in {4,5}$, every projective planar $r$-graph with no Petersen-minor is $r$-edge colorable.