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

Abstract

Let G be a simple finite graph and G ′ be a subgraph of G. A G ′ -design (X, B) of order n is said to be embedded into a G-design (X ∪ U, C) of order n + u, if there is an injective function f : B → C such that B is a subgraph of f (B) for every B ∈ B. The function f is called an embedding of (X, B) into (X ∪ U, C). If u attains the minimum possible value, then f is a minimum embedding. Here, by means of König'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).