Seurat games on Stockmeyer graphs

TL;DR

This paper introduces vertex colouring games on graphs and digraphs related to algebraic logic and reconstruction conjectures, showing their ability to distinguish non-isomorphic structures beyond existing algorithms and linking them to longstanding open problems.

Contribution

It establishes a connection between vertex colouring games and reconstruction conjectures, demonstrating their power to distinguish certain non-isomorphic graphs and digraphs, and relates these games to algebraic logic problems.

Findings

2-colour game distinguishes certain non-isomorphic graphs beyond Weisfeiler-Leman.

The game relates to and supports the reconstruction conjecture.

It can differentiate graphs in Stockmeyer's counterexample families.

Abstract

We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number $n$ such that $\forall$ always has a winning strategy in the game with $n$ colours whenever $G\not\cong H$. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with $n=3$ for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any $k<\omega$) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the $k$-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture.