Arithmetical rank of squarefree monomial ideals generated by five elements or with arithmetic degree four

Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $\mu (I) \leq 5$; (2) $\arithdeg I \leq 4$.