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

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2‐factorization of Kv∗ into α CM‐factors and β CN‐factors, with a Cℓ‐factor of Kv∗ being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd v≠MNgcd(M,N) the above necessary conditions are sufficient, except possibly when α=1, or when β∈{1,3}. Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

“…In the nonuniform case, it is known that OP( F ) has no solution if

F \in {false[3^{2} false], false[3^{4} false], false[4, 5 false], false[3^{2}, 5 false]}

, but that there is a solution for every other instance in which

v \leq 40

Section: Introductionmentioning

confidence: 99%

On the Hamilton–Waterloo problem with odd cycle lengths

Burgess

Danziger

Traetta

2017

“…Application In , the authors found a method to construct

1

‐rotational solutions to many different Oberwolfach problems. In particular, it is shown that for all

α \geq 1

and

k \geq 3

there exists a

1

‐rotational solution to

O P (2^{α} + 1,^{2} k,^{2^{α} - 1} (2 k))

when

k

is odd and

O P (2^{α} + 1,^{2^{α} + 1} k)

when

k

is even (Proposition 3.9 in ).…”

Section: Using a 2‐starter Of Kbold-italicgtrue¯ To Obtain A 2n‐startmentioning

confidence: 99%

“…Application In , the authors found a method to construct

1

‐rotational solutions to many different Oberwolfach problems. In particular, it is shown that for all

α \geq 1

and

k \geq 3

there exists a

1

‐rotational solution to

O P (2^{α} + 1,^{2} k,^{2^{α} - 1} (2 k))

when

k

is odd and

O P (2^{α} + 1,^{2^{α} + 1} k)

when

k

is even (Proposition 3.9 in ). Therefore we obtain solutions to

O P (p 2^{α} + 1,^{2 p} k,^{p 2^{α} - 1} (2 k))

for each odd prime

p

and

O P (p 2^{α} + 1,^{p (2^{α} + 1)} k)

for all primes

p

after applying Theorem with the same conditions on

α

and

k

.…”

Section: Using a 2‐starter Of Kbold-italicgtrue¯ To Obtain A 2n‐startmentioning

confidence: 99%

A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem

McGinnis

Poimenidou

2018

The concept of a 1‐rotational factorization of a complete graph under a finite group G was studied in detail by Buratti and Rinaldi. They found that if G admits a 1‐rotational 2‐factorization, then the involutions of G are pairwise conjugate. We extend their result by showing that if a finite group G admits a 1‐rotational k‐factorization with k = 2 n m even and m odd, then G has at most m ( 2 n − 1 ) conjugacy classes containing involutions. Also, we show that if G has exactly m ( 2 n − 1 ) conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational 2 n‐factorization under G × Z n given a 1‐rotational 2‐factorization under a finite group G. This construction, given a 1‐rotational solution to the Oberwolfach problem O P ( a ∞ , a 1 , a 20.05em … , a n ), allows us to find a solution to O P ( 2 a ∞ − 1 , 2-0.2em a 1 , 2-0.2em a 20.05em … , 2-0.2em a n ) when the a i’s are even ( i ≠ ∞), and O P ( p ( a ∞ − 1 ) + 1 , p-0.2em a 1 , p-0.2em a 20.05em … , p-0.15em a n ) when p is an odd prime, with no restrictions on the a i’s.

“…First note that if

T = [g_{1}, g_{2}, \dots, g_{2 n}] \in {scriptT}_{b s} (G)

, then

\bar{T} = [{\bar{g}}_{1}, {\bar{g}}_{2}, \dots, {\bar{g}}_{n}] \in {scriptT}_{b} (\bar{G})

. We need the following result due to Anderson whose proof will be sketched below following the doubling construction for Oberwolfach problems described in (recall that a HCS

(2 n + 1)

is nothing but a solution of the Oberwolfach problem OP

(2 n + 1)

). Theorem The counterimage of any basic terrace of

\bar{G}

under the map

π : [g_{1}, g_{2}, \dots, g_{2 n}] \in {scriptT}_{b s} (G)  [{\bar{g}}_{1}, {\bar{g}}_{2}, \dots, {\bar{g}}_{n}] \in T_{b} (\bar{G}),

has size

2^{(n + a (\bar{G}) - 1) / 2}

, where

a (\bar{G})

is the number of involutions of

\bar{G}

. Proof Let

U = [{\bar{g}}_{1}, {\bar{g}}_{2}, \dots, {\bar{g}}_{n}]

be any given terrace of

{scriptT}_{b} (\bar{G})

.…”

Section: Enumerating 1‐rotational Hamiltonian Cycle Systems Up To Isomentioning

confidence: 99%

“…We need the following result due to Anderson [4] whose proof will be sketched below following the doubling construction for Oberwolfach problems described in [21] (recall that a HCS(2n + 1) is nothing but a solution of the Oberwolfach problem OP(2n + 1)).…”

Section: Enumerating 1-rotational Hamiltonian Cycle Systems Up To Isomentioning

confidence: 99%

Some Results on 1‐Rotational Hamiltonian Cycle Systems

Buratti

Rinaldi

Traetta

2013

Self Cite

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