Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal I t (Γ ) is generated by monomials that correspond to directed paths of length (t − 1) in Γ . In this paper, we study algebraic properties and invariants of I t (Γ ). We give a recursive formula to compute the graded Betti numbers of I t (Γ ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when I t (Γ ) has a linear resolution.

a b s t r a c tWe introduce a conjecture about constructing critically (s + 1)-chromatic graphs from critically s-chromatic graphs. We then show how this conjecture implies that any unmixed height two square-free monomial ideal I in a polynomial ring R, i.e., the cover ideal of a finite simple graph, has the persistence property, that is, Ass(R/I s ) ⊆ Ass(R/I s+1 ) for all s ≥ 1. To support our conjecture, we prove that the statement is true if we also assume that χ f (G), the fractional chromatic number of the graph G, satisfies χ (G) − 1 < χ f (G) ≤ χ (G). We give an algebraic proof of this result.

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