  Let $G$ be a graph. A total dominating set in a graph $G$ is a set $S$ of
vertices of $G$ such that every vertex in $G$ is adjacent to a vertex in $S$.
Recently, the following question was proposed: "Is it true that every connected
cubic graph containing a $3$-cycle has two vertex disjoint total dominating
sets?" In this paper, we give a negative answer to this question. Moreover, we
prove that if we replace $3$-cycle with $4$-cycle the answer is affirmative.
This implies every connected cubic graph containing a diamond (the complete
graph of order $4$ minus one edge) as a subgraph can be partitioned into two
total dominating sets, a result that was proved in 2017.
