Altitude of regular graphs with girth at least five

The altitude of a graph G is the largest integer k such that for each linear ordering f of its edges, G has a (simple) path P of length k for which f increases along the edge sequence of P. We determine a necessary and sufficient condition for cubic graphs with girth at least five to have altitude three and show that for r⩾4, r-regular graphs with girth at least five have altitude at least four. Using this result we show that some snarks, including all but one of the Blanus˘a type snarks, have altitude three while others, including the flower snarks, have altitude four. We construct an infinite class of 4-regular graphs with altitude four.