Cup Stacking in Graphs

TL;DR

This paper introduces a new graph game called cup stacking, characterizes its stackability across various graph families, and develops algorithms to determine stackability, with implications for understanding graph structures and properties.

Contribution

It provides a comprehensive characterization of stackability in graphs, proves stackability for many graph families, and devises polynomial algorithms for diameter two graphs.

Findings

Complete graphs, paths, cycles, grids, Petersen graph, Kneser graphs, some trees, and hypercubes up to dimension 20 are stackable.

A polynomial algorithm is developed to decide stackability in diameter two graphs.

Cubes up to dimension 20 are proven to be stackable using combinatorial decompositions.

Abstract

Here we introduce a new game on graphs, called cup stacking, following a line of what can be considered as $0$-, $1$-, or $2$-person games such as chip firing, percolation, graph burning, zero forcing, cops and robbers, graph pebbling, and graph pegging, among others. It can be more general, but the most basic scenario begins with a single cup on each vertex of a graph. (This simplification coincides with an earlier game devised by Gordon Hamilton.) For a vertex with $k$ cups on it we can move all its cups to a vertex at distance $k$ from it, provided the second vertex already has at least one cup on it. The object is to stack all cups onto some pre-described target vertex. We say that a graph is stackable if this can be accomplished for all possible target vertices. In this paper we study cup stacking on many families of graphs, developing a characterization of stackability in graphs and using it to prove the stackability of complete graphs, paths, cycles, grids, the Petersen graph, many Kneser graphs, some trees, cubes of dimension up to 20, "somewhat balanced" complete $t$-partite graphs, and Hamiltonian diameter two graphs. Additionally we use the Gallai-Edmonds Structure Theorem, the Edmonds Blossom Algorithm, and the Hungarian algorithm to devise a polynomial algorithm to decide if a diameter two graph is stackable. Our proof that cubes up to dimension 20 are stackable uses Kleitman's Symmetric Chain Decomposition and the new result of Merino, M\"utze, and Namrata that all generalized Johnson graphs (excluding the Petersen graph) are Hamiltonian. We conjecture that all cubes and higher-dimensional grids are stackable, and leave the reader with several open problems, questions, and generalizations.