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If C is a clutter with n vertices and q edges whose clutter matrix has columnvectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is aHilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able toshow that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, thenC is an Ehrhart clutter and in this case we provide sharp bounds on theCastelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejolsconjecture on packing problems, we conjecture that if C is both ideal and theclique clutter of a perfect graph, then C has the MFMC property. We prove thisconjecture for Meyniel graphs, by showing that the clique clutters of Meynielgraphs are Ehrhart clutters. In much the same spirit, we provide a simple proofof our conjecture when C is a uniform clique clutter of a perfect graph. Weclose with a generalization of Ehrhart clutters as it relates to total dualintegrality.

Ehrhart clutters: Regularity and Max-Flow Min-Cut