I have the following question. It seems likely to be true - can anyone provide a standard reference?

Given: A connected, undirected graph.

Question 1: Can we assume a single direction for each edge such that the resulting directed graph is acyclic and strongly connected (path exists in one direction)?

Question 2: Suppose that if $i< j$ and there is an edge between $i$ and $j$ in the undirected graph, the edge in the directed graph is pointing from $i$ to $j$. Is there always a node numbering such that the resulting graph is strongly connected and acyclic? (e.g. would numbering down the branches of a spanning tree in the undirected graph work?)

Strongly connected(for a directed graph) usually means that between any two vertices there exist directed paths from one to the other; frequently, this is calleddiconnected. A possible counter-example (if I've understood the question correctly) is the edge and vertex set of the unit cube. – David Handelman Jul 3 at 11:19somelinear order. – Joel David Hamkins Jul 3 at 15:29