Title: Combinatorial properties of the G-degree,
Abstract: A strong interaction is known to exist between edge-colored graphs (which
encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as
a possible approach to the study of Quantum Gravity). The key tool is the {\it
G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the
tensor models context. In the present paper - by making use of combinatorial
properties concerning Hamiltonian decompositions of the complete graph - we
prove that, in any even dimension $d\ge 4$, the G-degree of all bipartite
graphs, as well as of all (bipartite or non-bipartite) graphs representing
singular manifolds, is an integer multiple of $(d-1)!$. As a consequence, in
even dimension, the terms of the $1/N$ expansion corresponding to odd powers of
$1/N$ are null in the complex context, and do not involve colored graphs
representing singular manifolds in the real context.
In particular, in the 4-dimensional case, where the G-degree is shown to
depend only on the regular genera with respect to an arbitrary pair of
"associated" cyclic permutations, several results are obtained, relating the
G-degree or the regular genus of 5-colored graphs and the Euler characteristic
of the associated PL 4-manifolds.