On Pauling's residual entropy estimate for regular graphs with growing degree

Abstract: In 1935, Pauling proposed an estimate for the number of Eulerian orientations of a graph in the context of the theoretical behaviour of water ice. The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is called the residual entropy. In an earlier paper, we conjectured that the residual entropy of a sequence of regular graphs of increasing degree was asymptotically equal to Pauling's estimate. Here we prove the conjecture under constraints on the number of short circuits. These constraints hold under weak eigenvalue conditions and apply to sequences of increasing girth and repeated Cartesian products such as hypercubes.