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

TL;DR

This paper provides a complete solution to the directed Oberwolfach problem for complete symmetric digraphs of order 2 mod 4, focusing on directed 2-factors with cycles of even lengths, advancing understanding of directed graph decompositions.

Contribution

It offers a novel, comprehensive solution to a specific case of the directed Oberwolfach problem involving cycles of even length for orders 2 mod 4.

Findings

Solved the directed Oberwolfach problem for order 2 mod 4 with even cycle lengths

Established existence of directed 2-factorizations in the specified case

Extended classical results to directed graph scenarios

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.