Elsevier

Discrete Mathematics

Volume 312, Issue 3, 6 February 2012, Pages 517-523
Discrete Mathematics

Exact embedding of two G-designs into a (G+e)-design

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Abstract

Let G be a connected simple graph and let SG be the spectrum of integers v for which there exists a G-design of order v. Put e={x,y}, with xV(G) and yV(G). Denote by G+e the graph having vertex set V(G){y} and edge set E(G){e}. Let (X,D) be a (G+e)-design. We say that two G-designs (Vi,Bi), i=1,2, are exactly embedded into (X,D) if X=V1V2, |V1V2|=0 and there is a bijective mapping f:B1B2D such that B is a subgraph of f(B), for every BB1B2. We give necessary and sufficient conditions so that two G-designs can be exactly embedded into a (G+e)-design. We also consider the following two problems: (1) determine the pairs {v1,v2}SG for which any two nontrivial G-designs (Vi,Bi), |Vi|=vi, i=1,2, can be exactly embedded into a (G+e)-design; (2) determine the pairs {v1,v2}SG for which there exists a (G+e)-design of order v1+v2 exactly embedding two nontrivial G-designs (Vi,Bi), |Vi|=vi, i=1,2. We study these problems for BIBDs, cycle systems, cube systems, path designs and star designs.

Keywords

Embedding
G-design
T-balanced

Cited by (0)

Supported by MIUR and by C.N.R. (G.N.S.A.G.A.), Italy.