On a Well-Spread Halving of Directed Multigraphs

Abstract

Let $G$ be an even degree multigraph and let $deg(v)$ and $p(uv,G)$ denote the degree of vertex $v$ in $G$ and the multiplicity of edge $(u,v)$ respectively in $G$. A decomposition of $G$ into multigraphs $G_1$ and $G_2$ is said to be a $well-spreadhalving$ of $G$ into two halves $G_1$ and $G_2$, if for each vertex $v$, $deg(v,G_1)=deg(v,G_2)=\frac{1}{2}deg(v,G)$, and $|\mu (uv,G_1)–\mu (uv,G_2)|\le 1$ for each edge $(u,v)\in E(G)$. A sufficient condition was given in $[7]$ under which there exists a well-spread halving of $G$ if we allow the addition/removal of a Hamilton cycle to/from $G$. Analogous to $[7]$, in this paper we define a well-spread halving of a directed multigraph $D$ and give a sufficient condition under which there exists a well-spread halving of $D$ if we allow the addition/removal of a particular type of Hamilton cycle to/from $D$.