Abstract
If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X∈Γ. In this article we completely solve the existence problem for decompositions of Kv-I into Cn-factors and K1,n-factors in the case when n is even.
Highlights
Introduction and DefinitionsFor any graph G, let V ( G ) and E( G ) be the vertex-set and the edge-set of G, respectively.Throughout the paper Kv will denote the complete graph on v vertices, while Kv \ Kh will denote the graph with V (Kv ) as vertex-set and E(Kv ) \ E(Kh ) as edge-set.Given a set Γ of pairwise non-isomorphic graphs, a Γ-decompositionof a graph G is a decomposition of the edge-set of G into subgraphs that are isomorphic to some element of Γ
If Γ is the set of all possible cycles of Kv, determining the existence of possible Γ-isofactorizations of Kv with an odd v is known as the Oberwolfach Problem
Construction 3. (Frame-Construction) Let F be a Γ-frame of type gu, where Γ is a set of graphs of order n ≥ 2 and the number of partial factors missing any fixed group is α, and let t, h and v be positive integers such that v = gtu + h
Summary
For any graph G, let V ( G ) and E( G ) be the vertex-set and the edge-set of G, respectively. If Γ is the set of all possible cycles of Kv , determining the existence of possible Γ-isofactorizations of Kv with an odd v is known as the Oberwolfach Problem. It was first posed in 1967 by Gerhard Ringel and asks whether it is possible to seat an odd number v of mathematicians at n round tables in (v − 1)/2 meals so that each mathematician sits next to everyone else exactly once. The uniform Oberwolfach problem (all cycles of a factor have the same size) has been completely solved by Alspach and Häggkvist [3]. There exists a (Cn , K1,n )-URD∗ (v; r, s) if and only if (r, s) ∈ J (v)
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