Minimum embedding of any Steiner triple system into a 3-sun system via matchings

TL;DR

This paper provides a complete solution for embedding any Steiner Triple System into a 3-sun system with minimal additional vertices, using edge coloring techniques and König's Line Coloring Theorem.

Contribution

It introduces a method to determine the minimum embedding of any Steiner Triple System into a 3-sun system, expanding understanding of graph design embeddings.

Findings

Established a minimum embedding framework for STS into 3-sun systems.

Applied König's Line Coloring Theorem to solve embedding problems.

Provided explicit bounds and constructions for minimal embeddings.

Abstract

Let $G$ be a simple finite graph and $G'$ be a subgraph of $G$. A $G'$-design $(X,\cal B)$ of order $n$ is said to be embedded into a $G$-design $(X\cup U,\cal C)$ of order $n+u$, if there is an injective function $f:\cal B\rightarrow \cal C$ such that $B$ is a subgraph of $f(B)$ for every $B\in\cal B$. The function $f$ is called an embedding of $(X,\cal B)$ into $(X\cup U,\cal C)$. If $u$ attains the minimum possible value, then $f$ is a minimum embedding. Here, by means of K\"{o}nig's Line Coloring Theorem and edge coloring properties a complete solution is given to the problem of determining a minimum embedding of any $K_3$-design (well-known as Steiner Triple System or, shortly, STS) into a 3-sun system or, shortly, a 3SS (i.e., a $G$-design where $G$ is a graph on six vertices consisting of a triangle with three pendant edges which form a 1-factor).