Harmonic Labeling of Graphs

TL;DR

This paper investigates which graphs can be labeled with integers via harmonic functions that are both injective and surjective, establishing existence results for grid graphs and non-existence for certain product graphs.

Contribution

It introduces the concept of harmonic labeling of graphs, constructs such labelings for the ^2 grid, and proves non-existence for product graphs like G   Z.

Findings

Harmonic labeling exists for the ^2 grid.

No harmonic labeling exists for G  Z when G has at least one edge.

Provides criteria for the existence of harmonic labelings on product graphs.

Abstract

Which graphs admit an integer value harmonic function which is injective and surjective onto $\Z$? Such a function, which we call harmonic labeling, is constructed when the graph is the $\Z^2$ square grid. It is shown that for any finite graph $G$ containing at least one edge, there is no harmonic labeling of $ G \times \Z$.