A note on the existence of an alternating sign on a spanning tree of   graphs
  For a spanning tree T of a connected graph G and for a labelling \phi: E(T) \rightarrow {+, -}, \phi is called an alternating sign on a spanning tree T of a graph G if for any cotree edge e \in E(G)-E(T), the unique path in T joining both end vertices of e has alternating signs. In the present note, we prove that any graph has a spanning tree T and an alternating sign on T. 
Dongseok Kim, Young Soo Kwon and Jaeun Lee
math.CO math.GT