On 2-cycles of graphs

Abstract

Let G=(V,E) be a finite undirected graph. Orient the edges of G in an arbitrary way. A 2-cycle on G is a function d:E2→Z such for each edge e, d(e,⋅) and d(⋅,e) are circulations on G, and d(e,f)=0 whenever e and f have a common vertex. We show that each 2-cycle is a sum of three special types of 2-cycles: cycle-pair 2-cycles, Kuratowski 2-cycles, and quad 2-cycles. In the case that the graph is Kuratowski connected, we show that each 2-cycle is a sum of cycle-pair 2-cycles and at most one Kuratowski 2-cycle. Furthermore, if the graph is Kuratowski connected, we characterize when every Kuratowski 2-cycle is a sum of cycle-pair 2-cycles. A consequence of this is that if G is Kuratowski connected and either G is planar or G does not have a linkless embedding, then each 2-cycle on G is a sum of cycle-pair 2-cycles. A 2-cycle d on G is skew-symmetric if d(e,f)=−d(f,e) for all edges e,f∈E. We show that each skew-symmetric 2-cycle is a sum of two special types of skew-symmetric 2-cycles: skew-symmetric cycle-pair 2-cycles and skew-symmetric quad 2-cycles. In the case that the graph is Kuratowski connected, we show that each skew-symmetric 2-cycle is a sum of skew-symmetric cycle-pair 2-cycles. Similar results like this had previously been obtained by one of the authors for symmetric 2-cycles. Symmetric 2-cycles are 2-cycles d such that d(e,f)=d(f,e) for all edges e,f∈E.