Sum index, difference index and exclusive sum number of graphs

TL;DR

This paper investigates conjectures relating sum index, difference index, and exclusive sum number of graphs, disproving some and providing explicit constructions and bounds to clarify their relationships.

Contribution

The paper refutes a conjectured exact relationship between sum and difference indices, offers new bounds, and constructs explicit examples showing the potential for large discrepancies.

Findings

The predicted relationship between sum and difference index can be arbitrarily inaccurate.

Explicit constructions demonstrate the exclusive sum number can greatly exceed the sum index.

New bounds are established for both sum and difference indices.

Abstract

We consider two recent conjectures of Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index and exclusive sum number of graphs. One conjecture posits an exact relationship between the first two invariants; we show that in fact the predicted value may be arbitrarily far from the truth in either direction. In the process we establish some new bounds on both the sum and difference index. The other conjecture, that the exclusive sum number can exceed the sum index by an arbitrarily large amount, follows from known, but non-constructive, results; we give an explicit construction demonstrating it. Simultaneously with the first version of this paper appearing, Harrington et al. updated their preprint with two counterexamples to the first conjecture; however, their counterexamples only give a discrepancy of 1, and only in one direction. They therefore modified the conjecture from an equality to an inequality; our results show that this is still false in general.