Down-linking $(K_v,\Gamma)$-designs to $P_3$-designs

TL;DR

This paper introduces the concept of down-linking in graph designs, proving that such transformations are generally possible and providing bounds for specific cases like paths of length three.

Contribution

It defines the new concept of down-linking in graph designs and establishes conditions under which such links exist, including explicit bounds and constructions.

Findings

Any (K_v,G)-design can be down-linked to a (K_n,G')-design if n is large enough.

For G'=P_3, a down-link exists to a design of order at most v+3.

Bounds are improved for specific graph classes using explicit constructions.

Abstract

Let G' be a subgraph of a graph G. We define a down-link from a (K_v,G)-design B to a (K_n,G')-design B' as a map f:B->B' mapping any block of B into one of its subgraphs. This is a new concept, closely related with both the notion of metamorphosis and that of embedding. In the present paper we study down-links in general and prove that any (K_v,G)-design might be down-linked to a (K_n,G')-design, provided that n is admissible and large enough. We also show that if G'=P_3, it is always possible to find a down-link to a design of order at most v+3. This bound is then improved for several classes of graphs Gamma, by providing explicit constructions.