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, some results on the embedding of Ck-systems into k-sun systems are obtained and a complete solution to the problem of determining a minimum embedding of any Steiner Triple System into a 3-sun system is given.