The regular number of a graph

Let $G$ be a simple undirected graph. The regular number of $G$ is defined to be the minimum number of subsets into which the edge set of $G$ can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain the regular number of some families of graphs and discuss some general bounds on this parameter. Also, some of the lower or upper bounds proved in \cite{Kulli:Janakiram:Iyer:2001} are shown here to hold with equality.

💡 Research Summary

The paper introduces and systematically studies a new graph invariant called the regular number (r(G)). For a simple undirected graph (G=(V,E)), (r(G)) is defined as the smallest integer (k) such that the edge set (E) can be partitioned into (k) subsets (E_1,\dots,E_k) and each induced subgraph (G