Monotonic Arrangements of Undirected Graphs

The problem of finding a path in a network each edge of which is at least as long as the previous edge has attracted some attention in recent years. For example, problem 10 of the 2008 American Invitational Mathematics Exam asked a variation of this problem, namely to find the number of paths of maximal length in the 4× 4 rectangular grid of dots such that the edge length strictly increases from beginning to end. See [2]. Also, see [1]. A collection {a1, a2, a3, · · · , an} of n ≥ 2 distinct points in the plane is said to be monotone if a1, a2, · · · , an can be ordered in some way ai1, ai2, · · · , ain such that the sequence of consecutive distances D (ai1, ai2) , D (ai2, ai3) , · · · , D ( ain−1 , ain ) is non-decreasing. That is, D (ai1, ai2) ≤ D (ai2, ai3) ≤ · · · ≤ D ( ain−1 , ain ) . In this note, we show that n ≥ 2 distinct points in the plane is always monotone if and only if n = 2, 3. We also discuss the same problem for the general abstract graph where the ( n 2 ) edges of a complete undirected graph on n ≥ 2 vertices are assigned arbitrary real numbers. We show that this abstract graph can always be monotonically arranged for a binary graph where we assign just two different real numbers d,D, d < D, to the ( n 2 ) edges. We then generalize this binary theorem in a very primitive way to give necessary and sufficient conditions so that any given abstract graph is monotone. The ultimate goal is to create a theorem analogous to Hall’s marriage theorem that gives a more reasonable solution to this problem. We then