Graphs and Degrees · Hard problem

A network planner proposes a list of degrees for $12$ computers: five of them listed as degree $3$ and the other seven as degree $4$. Exactly $5$ of the $12$ proposed degrees are odd. By the handshake lemma's corollary (the number of odd-degree vertices in any simple graph is always even), this list cannot be realized. What is the SMALLEST number of entries in the list that must be changed by exactly $1$ (up or down) to remove this obstruction?