# The Interactive Sum Choice Number of Graphs

## TL;DR

This paper introduces the interactive sum choice number, a new variant of graph coloring that adaptively assigns colors to vertices, and provides evidence that it is generally smaller than the traditional sum choice number.

## Contribution

It defines the interactive sum choice number, explores its properties, and demonstrates it is often strictly less than the sum choice number for various graph classes.

## Key findings

- The interactive sum choice number is always less than or equal to the sum choice number.
- For many graph classes, the difference between the two grows linearly with the number of vertices.
- Evidence supports the conjecture that the interactive sum choice number is strictly smaller than the sum choice number, except for complete graphs.

## Abstract

We introduce a variant of the well-studied sum choice number of graphs, which we call the interactive sum choice number. In this variant, we request colours to be added to the vertices' colour-lists one at a time, and so we are able to make use of information about the colours assigned so far to determine our future choices. The interactive sum choice number cannot exceed the sum choice number and we conjecture that, except in the case of complete graphs, the interactive sum choice number is always strictly smaller than the sum choice number. In this paper we provide evidence in support of this conjecture, demonstrating that it holds for a number of graph classes, and indeed that in many cases the difference between the two quantities grows as a linear function of the number of vertices.