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A complete solution to the directed Oberwolfach problem of order $2 \pmod{4}$ with cycles of even lengths

A. C. Burgess, P. H. Danziger, A. Lacaze-Masmonteil

Abstract

The Oberwolfach problem asks for a $2$-factorization of the complete graph in which each $2$-factor is isomorphic to a specific factor $F$. Recently, this problem has been extended to directed graphs. In this case, the directed Oberwolfach problem asks for a directed 2-factorization of the complete symmetric digraph in which each directed $2$-factor is isomorphic to a specific directed factor $F$. In this paper, we consider the directed Oberwolfach problem with directed 2-factors comprised of cycles of even lengths. Specifically, we provide a complete solution to this particular case when the order of the complete symmetric digraph is congruent to 2 modulo 4.

A complete solution to the directed Oberwolfach problem of order $2 \pmod{4}$ with cycles of even lengths

Abstract

The Oberwolfach problem asks for a -factorization of the complete graph in which each -factor is isomorphic to a specific factor . Recently, this problem has been extended to directed graphs. In this case, the directed Oberwolfach problem asks for a directed 2-factorization of the complete symmetric digraph in which each directed -factor is isomorphic to a specific directed factor . In this paper, we consider the directed Oberwolfach problem with directed 2-factors comprised of cycles of even lengths. Specifically, we provide a complete solution to this particular case when the order of the complete symmetric digraph is congruent to 2 modulo 4.
Paper Structure (10 sections, 20 theorems, 29 equations, 7 figures)

This paper contains 10 sections, 20 theorems, 29 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Setup and approach
  4. Factorizations of $W_{2m}^*$
  5. Acknowledgements
  6. Conflict of Interest Statement
  7. An admissible $[4,8]$-decomposition of $J_{12}^*$
  8. Caps and centre piece for a $[4,2t]$-decomposition of $J_{2t+4}^*$, $t \equiv 1 \pmod{4}$
  9. Right caps for Lemma \ref{['GeneralFactors']}
  10. Admissible decompositions of $J_{2m}^*$ with small cycle lengths

Key Result

Theorem 1.1

Let $m \geq 2$ be an integer. There exists a solution to $\mathop{\mathrm{OP}}\nolimits^*([m^{\alpha}])$ if and only if $(m,\alpha) \notin \{(4,1),(6,1), (3,2)\}$.

Figures (7)

  • Figure 1: The underlying graphs of $H^*_{14}$ and $W^*_{14}$. Vertices with the same label are identified.
  • Figure 2: The graph $J_{14}$.
  • Figure 3: Admissible $2$-regular subdigraphs in $J^*_8$ and $J^*_6$ that saturate the same vertices in $\{x_0, x_1, y_0, y_1\}$.
  • Figure 4: An admissible $[2,6,6]$-digraph of $J^*_{14}$ formed from the pair of admissible decompositions given in Figures \ref{['Fig:AdmiJ8']} and \ref{['Fig:AdmiJ6']} as described in Example \ref{['ex:comp']}.
  • Figure 5: Four directed paths used to construct an admissible $[2m]$-factor of $J^*_{2m}$, as described in Example \ref{['ex:capsplice']}.
  • ...and 2 more figures

Theorems & Definitions (47)

  • Theorem 1.1: AbelAdaBryBenZhaBerGerSotBurFranSajBurSajAliceTil
  • Theorem 1.2: ZanDu
  • Theorem 1.3: DanielAliceKadriSajna
  • Theorem 1.4: KadriSajnaQui
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1: DanielAlice
  • Lemma 3.2: Haggkvist
  • ...and 37 more