Some Cyclic Solutions to the Three Table Oberwolfach Problem

Abstract

We use graceful labellings of paths to give a new way of constructing terraces for cyclic groups. These terraces are then used to find cyclic solutions to the three table Oberwolfach problem, ${\rm OP}(r,r,s)$, where two of the tables have equal size. In particular we show that, for every odd $r \geq 3$ and even $r$ with $4 \leq r \leq 16$, there is a number $N_r$ such that there is a cyclic solution to ${\rm OP}(r,r,s)$ whenever $s \geq N_r$. The terraces we are able to construct also prove a conjecture of Anderson: For all $m \geq 3$, there is a terrace of ${\Bbb Z}_{2m}$ which begins $0, 2k, k, \ldots$ for some $k$.

2‐Starters, Graceful Labelings, and a Doubling Construction for the Oberwolfach Problem

“…Check that it is, in particular, a twofold 2-starter of Z 22 . Applying the doubling construction to , we obtain the 2-starter of Z 44 whose cycles are (5,14,17,18,10,6,8,27,36,39,40,32,28,30), (9,16,21,4,20), (31,38,43,26,42), (∞, 0, 23,13,37,7,25,19,34,2,33,11,24,12,41,3,29,15,35,1,22), and hence a 1-rotational solution for OP (21,14,5,5). Since has pattern [11; 7; 5], by means of Theorem 4.3 we see that {2,20} , {6,16} , and {2,6,16,20} have patterns [11; −; 5, 7], [11; 5, 7; −], and [11; 5; 7], respectively.…”

“…In this section, we develop an interesting idea by Ollis used in [27,30]. There, graceful labelings of paths are used to construct terraces (of the cyclic group) with some "extra" properties that allow to get infinite solutions of OP(2 − 1, , ) upon the application of a "lift construction."…”

“…For instance, the well-known Walecki construction yields a 1-rotational solution to OP(2n + 1). In a series of papers [27][28][29][30], new solutions to the OP are provided (with a special attention to the case with three tables) and they are basically achieved via the concept of a "sectionable terrace" and its variants. All these solutions turn out to be 1-rotational over the cyclic group, although the authors speak of cyclic solutions.…”

“…The twofold OP is dealt with in [28], where 1-rotational solutions to OP (3, ), for any ≥ 4, and OP (3, 3, ), for any odd = 5, are given. In [27,30], Ollis and Willmott show that OP(r, r, s), with s odd, has a 1-rotational solution for any r ≥ 14 provided that s ≥ 5r/2 + 1 or r + 1 according to whether r is odd or even, respectively, and then solve completely the problem whenever 3 ≤ r ≤ 13 and s ≥ 5. In this way, they solve infinite cases left unsettled by Hilton and Johnson [24].…”

1‐rotational k‐factorizations of the complete graph and new solutions to the Oberwolfach problem

“…We remark that 1rotational solutions to OP( r 3) were constructed for some classes of values of r in [9,10]. We also remark that in [21,22] some examples of 1-rotational 2-factorizations under the action of a cyclic group are constructed via sectionable terraces. In particular, in [21] the following 1-rotational solutions are provided: OP (3, 3, k) for each odd k ≥ 7; OP(2k + 1, 2k, 2k) and OP(2k + 1, 2k + 1, 2k + 1) for every k ≥ 2.…”