Pauling residual entropy conjecture for growing-degree regular graphs

Establish that, for every sequence of regular graphs G(n) with even degree d(n) tending to infinity, the residual entropy rho(G(n)) is asymptotic to Pauling’s residual entropy estimate rho_P(G(n)), namely rho(G(n)) = rho_P(G(n)) + o(1).

Background

For a d-regular graph G with even d, the residual entropy is the logarithm of the number of Eulerian orientations of G divided by the number of vertices. Pauling’s estimate is obtained by treating the events that individual vertices have equal in-degree and out-degree as independent. Lieb and Wu established that the true residual entropy is at least Pauling’s estimate.

The conjecture asserts that the difference between the true residual entropy and Pauling’s estimate tends to zero for every sequence of regular graphs whose even degrees grow without bound. The paper proves this assertion under a restriction on the number of short closed trails, and derives consequences for graphs with suitable eigenvalue conditions, growing girth, and certain repeated Cartesian products. Thus, the unrestricted statement remains unresolved beyond the classes covered by the paper.