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 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.

Uniformly resolvable decompositions of Kv into paths on two, three and four vertices

“…Hence, D (12) and apply Theorem 3.5 with g = 6, u = v−12 24 , t = 12 and h = 12 (the input designs are: a (P 2 , P 3 , (12), which exists by Lemma 4.3; a (P 2 , P 3 , P 4 ) − URGDD(r ′′ 2 , r ′′ 3 , r ′′ 4 ) of type 12 2 with (r ′′ 2 , r ′′ 3 , r ′′ 4 ) ∈ D(12 2 ), which exists by Lemma 4.4; a (P 2 , P 3 , P 4 ) − IURD (36, 12; [r ′ 2 , r ′ 3 , r ′ 4 ], [r ′′′ 2 , r ′′′ 3 , r ′′′ 4 ]) with (r ′ 2 , r ′ 3 , r ′ 4 ) ∈ D(12) and (r ′′′ 2 , r ′′′ 3 , r ′′′ 4 ) ∈ 2 * D(12 2 ), which exists by Lemma 4.8). This implies URD(v; P 2 , P 3 , P 4 ) ⊇ D(12) + v − 12 24 *  2 * D(12 2 )  .…”

“…A variant of the Oberwolfach problem is known as Hamilton-Waterloo problem and asks for a 2-factorization of K v having an assigned number r of factors all isomorphic to an assigned 2-regular graph H and the remaining (v − 2r − 1)/2 factors all isomorphic to another assigned 2-regular graph W . The Oberwolfach problem has been solved for a single cycle size [3] and recently for the case where the factors contain exactly two cycles [35]; for solutions to the analogous Oberwolfach Problem for complete multipartite graphs see [6,7,23,24], and [28]. As regards the Hamilton-Waterloo problem, several cases have been solved in [2,11,12], and [18], where the authors have focused on the case where both H and W are uniform, i.e., the case where all cycles of H have an assigned length h and all cycles of W have an assigned length w; in particular, the case where h = 3 and w = 4 is solved in [11], with some exceptions which are solved in [5]; the case where h = 3 and w = v is studied in [12] and [18]; finally, many other solutions to the Oberwolfach problem and to the Hamilton-Waterloo problem can be found in the literature (see, for instance, [8] and [9]).…”

Cyclic Uniform 2-Factorizations of the Complete Multipartite Graph

“…Case 3: Let m = 10 and n = 6, hence t = 1. We obtain [12,18], [11,19], [10,20], [9,21], [8,22], [7,23], [6,24], [5,25], [4,26], [3,27], [2,28]…”

A note on the Hamilton–Waterloo problem with C8-factors and CmWang1, Cao2 2018Discrete Mathematics90100View full textAdd to dashboardCite

“…A factor of H is a spanning subgraph of H. Suppose G is a subgraph of H, a G-factor of H is a set of edge-disjoint subgraphs of H, each isomorphic to G. A G-factorization of H is a set of edge-disjoint G-factors of H. A C k -factorization of H is a partition of E(H) into C k -factors. Many papers introduced C k -factorization of K u [g], see [2,4,10,18,19,20,22,23]. Theorem 1.1.…”