Sprague-Grundy Function of Symmetric Hypergraphs

TL;DR

This paper characterizes symmetric hypergraphs in hypergraph NIM where the Sprague-Grundy function aligns with a known formula, extending understanding of combinatorial game theory for these structures.

Contribution

It provides a complete characterization of symmetric hypergraphs for which the Sprague-Grundy function matches the classical formula.

Findings

Identifies conditions under which the Sprague-Grundy function is given by the known formula.

Extends the class of hypergraphs with explicitly computable Sprague-Grundy functions.

Enhances understanding of the structure of symmetric hypergraphs in combinatorial game theory.

Abstract

We consider a generalization of the classical game of $NIM$ called hypergraph $NIM$. Given a hypergraph $\cH$ on the ground set $V = \{1, \ldots, n\}$ of $n$ piles of stones, two players alternate in choosing a hyperedge $H \in \cH$ and strictly decreasing all piles $i\in H$. The player who makes the last move is the winner. Recently it was shown that for many classes of hypergraphs the Sprague-Grundy function of the corresponding game is given by the formula introduced originally by Jenkyns and Mayberry (1980). In this paper we characterize symmetric hypergraphs for which the Sprague-Grundy function is described by the same formula.