On the Oberwolfach problem for single-flip $2$-factors via graceful labelings

TL;DR

This paper proves new constructive solutions for the Oberwolfach problem for certain 2-regular graphs with large cycles and single-flip automorphisms, using graceful labelings and explicit combinatorial constructions.

Contribution

It introduces a doubling construction method based on graceful labelings to explicitly solve the Oberwolfach problem for graphs with specific automorphisms and cycle length conditions.

Findings

Solutions exist for large cycle graphs with single-flip automorphisms.

Explicit constructions outperform probabilistic asymptotic methods.

Applicable to graphs with edges of multiplicity 2 without single-flip requirement.

Abstract

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem $OP(F)$, posed in 1967 and still open, asks for a decomposition of $K_v$ into copies of $F$. In this paper we show that $OP(F)$ has a solution whenever $F$ has a sufficiently large cycle which meets a given lower bound and, in addition, has a single-flip automorphism, which is an involutory automorphism acting as a reflection on exactly one of the cycles of $F$. Furthermore, we prove analogous results for the minimum covering version and the maximum packing version of the problem. We also show a similar result when the edges of $K_v$ have multiplicity 2, but in this case we do not require that $F$ be single-flip. Our approach allows us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms, in contrast with some recent asymptotic results, based on probabilistic methods, which are nonconstructive and do not provide a lower bound on the order of $F$ that guarantees the solvability of $OP(F)$. Our constructions are based on a doubling construction which applies to graceful labelings of $2$-regular graphs with a vertex removed. We show that this class of graphs is graceful as long as the length of the path-component is sufficiently large. A much better lower bound on the length of the path is given for an $\alpha$-labeling of such graphs to exist.