From graceful labellings of paths to cyclic solutions of the Oberwolfach problem

Every 1‐rotational solution of a classic or twofold Oberwolfach problem (OP) of order n is generated by a suitable 2‐factor (starter) of Kn or 2Kn, respectively. It is shown that any starter of a twofold OP of order n gives rise to a starter of a classic OP of order 2n−1 (doubling construction). It is also shown that by suitably modifying the starter of a classic OP, one may obtain starters of some other OPs of the same order but having different parameters. The combination of these two constructions leads to lots of new infinite classes of solvable OPs. Still more classes can be obtained with the help of a third construction making use of the possible gracefulness of a graph whose connected components are cycles and at most one path. As one of the many applications, Hilton and Johnson's [J London Math Soc, 64 (2001) 513–522] bound s≥5r−1 about the solvability of OP(r,s) is improved to s≥⌊r/4⌋+10 in the case of r even. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 483‐503, 2012

Section: From a 2-rotational Solution Of An Op Of Order 4n + 3 To 1-rmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Buratti

Traetta

2012

“…The terrace is said to be basic if one if its endpoints is the identity element of G ; it is directed if we have

{g_{i + 1} - g_{i} | 1 \leq i \leq n - 1} = G ∖ {0} .

In particular, the terrace T is certainly directed if there exists an involution

λ \in G

such that

g_{i} + λ

(or

g_{i} \cdot λ

) coincides with

g_{n + 1 - i}

for

1 \leq i \leq n

. In this special case T is said to be a symmetric directed terrace (see ) but, for the sake of brevity, we prefer to speak of a symmetric terrace .…”

Section: Introductionmentioning

confidence: 99%

“…Denote as usual with K v the complete graph on v vertices and with K v − I the complete graph on v vertices with one 1-factor I removed. A Hamiltonian cycle system of order In particular, the terrace T is certainly directed if there exists an involution λ ∈ G such that g i + λ (or g i · λ) coincides with g n+1−i for 1 ≤ i ≤ n. In this special case T is said to be a symmetric directed terrace (see [37]) but, for the sake of brevity, we prefer to speak of a symmetric terrace.…”

Section: Introductionmentioning

confidence: 99%

Some Results on 1‐Rotational Hamiltonian Cycle Systems

Buratti

Rinaldi

Traetta

2013

A Hamiltonian cycle system of the complete graph on v vertices (briefly, a HCS(v)) is 1-rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any integer n greatest or equal to 3, there exists a 3-perfect 1-rotational HCS(2n+1). This allows to get the existence of an infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion, a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, an imprtant lower bound for the number of non isomorphic 1-rotational (and hence symmetric) HCSs is obtained

“…There are numerous other results on the Oberwolfach Problem, dealing with various special families of 2-regular graphs, see [9,13,14,25,28,31,33,40,42,43,45,47] and see [11] for a survey of results up to…”

Section: Introductionmentioning

confidence: 99%

On the Hamilton‐Waterloo Problem for Bipartite 2‐Factors

Bryant

Danziger

Dean

2012

Given two 2-regular graphs F 1 and F 2 , both of order n, the Hamilton-Waterloo Problem for F 1 and F 2 asks for a factorisation of the complete graph K n into α 1 copies of F 1 , α 2 copies of F 2 , and a 1-factor if n is even, for all non-negative integers α 1 and α 2 satisfying α 1 + α 2 = n−1 2 . We settle the Hamilton-Waterloo problem for all bipartite 2-regular graphs F 1 and F 2 where F 1 can be obtained from F 2 by replacing each cycle with a bipartite 2-regular graph of the same order.