Navigating in a Graph by Aid of Its Spanning Tree Metric (original) (raw)

On zero-sum spanning trees and zero-sum connectivity

arXiv: Combinatorics, 2020

We consider 222-colourings f:E(G)rightarrow−1,1f : E(G) \rightarrow \{ -1 ,1 \}f:E(G)rightarrow−1,1 of the edges of a graph GGG with colours −1-1−1 and 111 in mathbbZ\mathbb{Z}mathbbZ. A subgraph HHH of GGG is said to be a zero-sum subgraph of GGG under fff if f(H):=sumeinE(H)f(e)=0f(H) := \sum_{e\in E(H)} f(e) =0f(H):=sumeinE(H)f(e)=0. We study the following type of questions, in several cases obtaining best possible results: Under which conditions on ∣f(G)∣|f(G)|∣f(G)∣ can we guarantee the existence of a zero-sum spanning tree of GGG? The types of GGG we consider are complete graphs, K3K_3K3-free graphs, ddd-trees, and maximal planar graphs. We also answer the question of when any such colouring contains a zero-sum spanning path or a zero-sum spanning tree of diameter at most 333, showing in passing that the diameter-$3$ condition is best possible. Finally, we give, for G=KnG = K_nG=Kn, a sharp bound on ∣f(Kn)∣|f(K_n)|∣f(Kn)∣ by which an interesting zero-sum connectivity property is forced, namely that any two vertices are joined by a zero-sum path of length at most 444. One feature of this paper is th...