Sum index, difference index and exclusive sum number of graphs

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.