We study the interacting self-avoiding trail (ISAT) model on a Bethe lattice of general coordination q and on a Husimi lattice built with squares and coordination q = 4. The exact grand-canonical solutions of the model are obtained, considering that up to K monomers can be placed on a site and associating a weight omega(i) with an i-fold visited site. Very rich phase diagrams are found with nonpolymerized, regular polymerized, and dense polymerized phases separated by lines (or surfaces) of continuous and discontinuous transitions. For a Bethe lattice with q = 4 and K = 2, the collapse transition is identified with a bicritical point and the collapsed phase is associated with the dense polymerized (solidlike) phase instead of the regular polymerized (liquidlike) phase. A similar result is found for the Husimi lattice, which may explain the difference between the collapse transition for ISATs and for interacting self-avoiding walks on the square lattice. For q = 6 and K = 3 (studied on the Bethe lattice only), a more complex phase diagram is found, with two critical planes and two coexistence surfaces, separated by two tricritical and two critical end-point lines meeting at a multicritical point. The mapping of the phase diagrams in the canonical ensemble is discussed and compared with simulational results for regular lattices.
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