Large Monotone Paths in Graphs with Bounded Degree

.   We prove that for every ε>0 and positive integer r, there exists Δ0=Δ0(ε) such that if Δ>Δ0 and n>n(Δ,ε,r) then there exists a packing of K n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn 2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K n ).