On ?-Fold Equipartite Oberwolfach Problem with Uniform Table Sizes

Abstract

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 T1, T2,...,Ts (where each Ti can accommodate ti ≥ 3 people and ∑ti = mn )f ork 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. Throughout the paper, we use Cn for a cycle on n vertices, Pn for a path on n vertices, Kn for the complete graph on n vertices, and K(m : n) for the complete n-partite graph with m vertices in each partite set (also called complete equipartite graph), namely, K(m : n )= K(m1, m2,...,mn) with m1 = m2 = ··· = mn = m. Also, for a graph G ,w e use λG to represent the multi-graph obtained from G by replacing each edge of G with λ copies of it. A fa ctor Fof a graph G is a subgraph for which V(F )= V(G) .A nr- fa ctorof G is a factor that is regular of degree r. Clearly, a 2-factor is a disjoint union of cycles. An r- f actorization of a graph G is a partition of the edge set E(G) into r- fa ctors. Thus, a graph G having a 2-factorization must be regular of even degree. An {H1, H2,...,Hk}- factorization of a graph G is a partition of the edge set E(G) into factors such that each component of any factor is isomorphic to Hi for some 1 ≤ i ≤ k. In particular, an H- f actorization of G is a partition of E(G) into factors such that each factor is a disjoint union of H's. Consequently, a Ct -factorization of G is a 2-factorization with each 2-factor being a disjoint union of Ct 's.