The depression of a graph and the diameter of its line graph

An edge ordering of a graph G = ( V , E ) is an injection f : E → Q + where Q + is the set of positive rational numbers. A (simple) path λ for which f increases along its edge sequence is an f -ascent, and a maximal f -ascent if it is not contained in a longer f -ascent. The depression ε ( G ) of G is the least integer k such that every edge ordering of G has a maximal ascent of length at most  k . been shown in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143–160] that the difference diam ( L ( G ) ) − ε ( G ) may be made arbitrarily large. We prove that the difference ε ( G ) − diam ( L ( G ) ) can also be arbitrarily large, thus answering a question raised in [E.J. Cockayne, G. Geldenhuys, P.J.P. Grobler, C.M. Mynhardt, J. van Vuuren, The depression of a graph, Utilitas Math. 69 (2006) 143–160].