is it always true that the depth of BFS is $\leq$ DFS?
A result about BFS is that what you call the depth of the BFS of a node $v$ is the length of a shortest path from the source to $v$.

Assume the exists an instance where the depth of the DFS $&lt;$ BFS. Then we have a path from the source to $v$ that has length strictly smaller (the path taken by the DFS to reach $v$) than the length of a shortest path from the source to $v$ (the path taken by the BFS), which is impossible. 

This argument by contradiction shows that depth BFS $\leq$ DFS, for any graph.