  The $4$-girth-thickness $\theta(4,G)$ of a graph $G$ is the minimum number of
planar subgraphs of girth at least four whose union is $G$. In this paper, we
obtain that the 4-girth-thickness of complete tripartite graph $K_{n,n,n}$ is
$\big\lceil\frac{n+1}{2}\big\rceil$ except for $\theta(4,K_{1,1,1})=2$. And we
also show that the $4$-girth-thickness of the complete graph $K_{10}$ is three
which disprove the conjecture $\theta(4,K_{10})=4$ posed by Rubio-Montiel (Ars
Math Contemp 14(2) (2018) 319).
