A note on the existence of an alternating sign on a spanning tree of graphs

TL;DR

This paper proves that every connected graph has a spanning tree with an alternating sign labeling, ensuring a specific sign pattern on the paths corresponding to cotree edges.

Contribution

It establishes the existence of a spanning tree with an alternating sign labeling for any connected graph, a new theoretical result.

Findings

Existence of such spanning trees for all connected graphs

The alternating sign property holds for any cotree edge

Provides a new perspective on graph labeling and spanning trees

Abstract

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.