On the complexity of deciding whether the regular number is at most two

The regular number of a graph G denoted by reg(G) is 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 answer to the problem posed as an open problem in A. Ganesan et al. (2012) [3] about the complexity of determining the regular number of graphs. We show that computation of the regular number for connected bipartite graphs is NP-hard. Furthermore, we show that, determining whether reg(G) = 2 for a given connected 3-colorable graph G is NP-complete. Also, we prove that a new variant of the Monotone Not-All-Equal 3-Sat problem is NP-complete.