On the principal edge bipartition of a graph

A cycle in a graph is a set of edges that covers each vertex an even number of times. An even cycle is cycle of even cardinality. A co cycle is a collection of edges that intersects each cycle in an even number of edges. A coeven is a co cycle or the complement of a cocycle. A bieven is a collection of edges that is both an even cycle and a coeven. The even cycles, coevens, and bievens each form a vector space over the integers modulo two when addition is defined as symmetric difference of sets. An edge is co even cyclic if it belongs to a co even C for which C {e} is an even cycle. An edge is bieven cyclic if it belongs to a bieven. We show that any edge in a graph is either coeven cyclic or bieven cyclic.