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 G-degree of the involved graphs, which drives the 1 / Nexpansion 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.
Edge-colored graph PL-manifold Singular manifold Colored tensor model Regular genus Gurau degree
Mathematics Subject Classification
57Q15 57N13 57M15 83E99
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The authors would like to thank Gloria Rinaldi for her helpful ideas and suggestions about relationship between cyclic permutation properties and Hamiltonian cycle decompositions of complete graphs. They also thank the referees for their very helpful suggestions.