The Existence of an Alternating Sign on a Spanning Tree of Graphs

For a spanning tree T of a connected graph ${\Gamma}$ and for a labelling ${\phi}$: E(T) ${\rightarrow}$ {+,-},${\phi}$ is called an alternating sign on a spanning tree T of a graph ${\Gamma}$ if for any cotree edge $e{\in}E({\Gamma})-E(T)$, the unique path in T joining both end vertices of e has alternating signs. In the present article, we prove that any graph has a spanning tree T and an alternating sign on T.