an:06137923
Zbl 1258.05099
Buratti, Marco; Traetta, Tommaso
2-starters, graceful labelings, and a doubling construction for the Oberwolfach problem
EN
J. Comb. Des. 20, No. 11-12, 483-503 (2012).
00310870
2012
j
05C70 05C78
Oberwolfach problem; 1-rotational 2-factorization; graceful labeling
Summary: Every 1-rotational solution of a classic or twofold Oberwolfach problem (OP) of order n is generated by a suitable 2-factor (starter) of \(K_n\) or \(2K_n\), 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 bound [\textit{A. J. W. Hilton} and \textit{M. Johnson}, J. Lond. Math. Soc., II. Ser. 64, No. 3, 513--522 (2001; Zbl 1012.05135)] \(s\geq 5r-1\) about the solvability of OP\((r,s)\) is improved to \(s\geq \lfloor r/4\rfloor+10\) in the case of \(r\) even.
Zbl 1012.05135