On nice graphs

A digraph G is k-nice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y which respects this pattern (pluses corresponding to forward edges and minuses to backward edges). A digraph is then nice if it is k-nice for some k. Similarly, a multigraph H, whose edges are coloured by a set of p colours, is k-nice if for every two (not necessarily distinct) vertices x and y in H and every pattern of length k, given as a sequence of colours, there exists a path of length k linking x to y which respects this pattern. Such a multigraph is nice if it is k-nice for some k. In this paper we study the structure of nice digraphs and multigraphs.