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

Abstract

Abstract. For a spanning tree T of a connected graph Γ and for a labelling φ : E(T) →{+,−}, φ is called an alternating sign on a spanning tree T of a graph Γ if for any cotreeedge e ∈ E(Γ)−E(T), the unique path in T joining both end vertices of e has alternatingsigns. In the present article, we prove that any graph has a spanning tree T and analternating sign on T. 1. IntroductionA graph Γ is an ordered pair Γ = (V (Γ),E(Γ)) comprising a set V(Γ) of verticestogether with a set E(Γ) of edges. A graph is signed if there is a function µ :E(Γ) → {+,−}. A labelling on Γ means to be a 2-edge coloring which is a functionφ : E(Γ) → {+,−} unless stated differently. A graph Γ is bipartite if vertices canbe divided into two disjoint sets S and T such that every edge connects a vertexin S to one in T. A tree is a connected acyclic simple graph. A spanning tree is aspanning subgraph that is a tree. A walk is an alternating sequence of vertices andedges, beginning and ending with a vertex, where each vertex is incident to boththe edge that precedes it and the edge that follows it in the sequence, and where thevertices that precede and follow an edge are the end vertices of that edge. A walkis closed if its first and last vertices are the same, and open if they are different. Apath is an open walk which is simple, meaning that no vertices (and thus no edges)are repeated. We will often omit edges in path if there is no ambiguity.For a spanning tree T of a connected graph Γ, a labeling φ : E(T) → {+,−} iscalled an alternating sign on a spanning tree T of a graph Γ if for any cotree edgee ∈ E(Γ)\E(T), the unique path v