Motivations and history of some of my conjectures

Let G be a (directed) graph with vertex set X and let k be a positive integer. A purtiul k-coloring (S,, S2, . . . ,S,) is a family of k disjoint stable sets of G. A vertex which belongs to Si is said to be ‘of color i’, but some vertices may be uncolored. We say that a partial k-coloring of G saturates a subset A of X if the number of difSerent colors represented in A is exactly B,(A) = min{k; IA]}, i.e. the maximum possible. A partition of X is saturated if each of its classes is saturated. A puth p can denote either a sequence of distinct vertices (directed elementary path) or the subset of X defined by these vertices. Consider a partition M = {p1, p2, 1 of X into paths; if k is a positive integer, we write

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