# New reducible configurations for graph multicoloring with application to the experimental resolution of McDiarmid-Reed's Conjecture (extended version)

## TL;DR

This paper introduces new reduction techniques for graph multicoloring problems, specifically for (a,b)-colorings, and applies these methods to computationally support McDiarmid-Reed's conjecture on triangle-free lattice subgraphs.

## Contribution

The paper develops general reduction tools for (a,b)-coloring of graphs within 2≤a/b≤3, and demonstrates their effectiveness on large graph datasets related to McDiarmid-Reed's conjecture.

## Key findings

- Successfully colored all but one graph in the dataset.
- Provided computational evidence supporting McDiarmid-Reed's conjecture.
- Developed reduction tools applicable to complex graph coloring problems.

## Abstract

A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le a/b\le 3$. In particular, using necessary and sufficient conditions for the existence of a $(a,b)$-coloring of a path with prescribed color-sets on its end-vertices, more complex $(a,b)$-colorability reductions are presented. The utility of these tools is exemplified on finite triangle-free induced subgraphs of the triangular lattice for which McDiarmid-Reed's conjecture asserts that they are all $(9,4)$-colorable. Computations on millions of such graphs generated randomly show that our tools allow to find a $(9,4)$-coloring for each of them except for one specific regular shape of graphs (that can be $(9,4)$-colored by an easy ad-hoc process). We thus obtain computational evidence towards the conjecture of McDiarmid\&Reed.