Suppose we have a simple directed graph and $m$ denotes the maximum between the minimum out-degree and minimum in-degree. Is there a path with length at least m?
I know this holds if $m$ denotes the minimum among those two, but for the maximum I am not sure.
The answer is positive. Indeed, pick any maximal path $P=(v_1,\dots,v_k)$ in the graph. If out-degree of $v_k$ is at least $m$ then maximality of $P$ implies that each vertex $u$ of the graph such that $(v_k,u)$ is an edge is one among $v_1,\dots, v_{k-1}$, so $m\le k-1$. The case when the in-degree of $v_1$ is at least $m$ is considered similarly. Thus $k\ge m+1$.