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 \({well-spread \;halving}\) 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 \(|\mu(uv, G_1) – \mu(uv, G_2)| \leq 1\) 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 \(D\).