Minimal Graph Embeddings via Point Deletions in Steiner triple systems

TL;DR

This paper explores minimal embeddings of various graphs into Steiner triple systems by analyzing point deletions, providing near-optimal constructions and insights into the structure of these embeddings.

Contribution

It introduces methods for embedding complete and star graphs into Steiner triple systems with minimal or near-minimal point deletions, advancing understanding of graph embeddings in combinatorial designs.

Findings

Embedded complete graphs and star graphs with minimal deletions

Near-optimal bounds for embeddings of certain graphs

Discussion of minimal embeddings for empty graphs, paths, and cycles

Abstract

The game Nofil is a two-player combinatorial game in which players take turns marking points of a design such that the set of marked points does not contain a block. Equivalently, we can think of the points as being deleted from the design and points that are on singleton sets can no longer be marked. Every game play eventually results in the design becoming a graph. Previous work has shown that every graph is reachable from some Steiner triple system (STS), although the order of the constructed STS is often far from the known lower bounds. In this paper we give embeddings of complete graphs and star graphs into a $\STS$ that is minimal or very nearly meets the bounds. We further discuss possible minimal embeddings of empty graphs, paths, and cycles.