Uniform coverings of 2-paths with 5-paths in the complete graph

Uniform coverings of 2-paths with 5-paths in K2n

We give a construction of a uniform covering of 2-paths with 5-paths in Kn for all even n ≥ 6, i.e., we construct a set S of 5-paths in Kn having the property that each 2-path in Kn lies in exactly one 5-path in S for all even n ≥ 6.

Uniform coverings of 2-paths in the complete graph and the complete bipartite graph

Let G be a graph and H a subgraph of G. A D(G,H, λ) design is a collection D of subgraphs of G each isomorphic to H so that every 2-path (path of length 2) in G lies in exactly λ subgraphs in D. The problem of constructing D(Kn, Cn, 1) designs is the so-called Dudeney’s round table problem. We denote by Ck a cycle on k vertices and by Pk a path on k vertices. In this paper, we construct D(Kn,n,...

Uniform coverings of 2-paths with 6-cycles in the complete graph

Uniform coverings of 2-paths in the complete bipartite directed graph

Let G be a directed graph and H a subgraph of G. A D(G,H, λ) design is a multiset D of subgraphs of G each isomorphic to H so that every directed 2-path in G lies in exactly λ subgraphs in D. In this paper, we show that there exists a D(K∗ n,n, −→ C 2n, 1) design for every n ≥ 2, where K∗ n,n is the complete bipartite directed graph and −→ C 2n is a directed Hamilton cycle in K∗ n,n.

Uniform coverings of 2-paths by 4-paths

We construct a uniform covering of 2-paths by 4-paths in Kn for all n 2:: 5, i.e., we construct a set S of 4-paths in Kn having the property that each 2-path in Kn lies in exactly one 4-path in S for all 11, 2:: 5.