Cycle systems in the complete bipartite graph minus a one-factor

AbstractLet Kn,n−I denote the complete bipartite graph with n vertices in each part from which a 1-factor I has been removed. An m-cycle system of Kn,n−I is a collection of m-cycles whose edges partition Kn,n−I. Necessary conditions for the existence of such an m-cycle system are that m⩾4 is even, n⩾3 is odd, m⩽2n, and m|n(n−1). In this paper, we show these necessary conditions are sufficient except possibly in the case that m≡0(mod4) with n<m<2n

A Characterization of Projective-Planar Signed Graphs

A signed graph has a plus or minus sign on each edge. A cycle is balanced or unbalanced depending on whether it contains an even or odd number of negative edges respectively. We consider embeddings of a signed graph in the projective plane where a cycle is essential if and only if it is unbalanced. We characterize those signed graphs that have such a projective planar embedding. Our characterization is in terms of balancing a related graph formed by considering the homeomorphs of K 2;3 in the given graph

Halin&apos;s Theorem for the Möbius Strip

Halin&apos;s Theorem characterizes those locally finite infinite graphs that embed in the plane without accumulation points by giving a set of six topologically-excluded subgraphs. We prove the analogous theorem for graphs that embed in an open Möbius strip without accumulation points. There are 153 such obstructions under the ray ordering defined herein. There are 350 obstructions under the minor ordering. There are 1225..