Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

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We consider the following generalization of the Oberwolfach problem: "At a gathering there are n delegations each having m people. Is it possible to arrange a seating of mn people present at s round tables T 1 , T 2 ,..., T s (where each T i can accommodate t i ≥ 3 people and ∑ti = mn) for k different meals so that each person has every other person not in the same delegation for a neighbor exactly λ times?" For λ = 1, Liu has obtained the complete solution to the problem when all tables accommodate the same number t of people. In this paper, we give the complete solution to the problem for λ ≥ 2 when all tables have uniform sizes t.

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