Asymptotic number of edge-colored regular graphs

TL;DR

The paper studies the asymptotic enumeration of edge-colored k-regular graphs with prescribed vertex-incidence structures under half-edge labeling and automorphism weighting. It builds an analytic framework around a homogeneous polynomial V of degree k and its associated g, derives a sphere-integral representation for the weighted count A(n), and uses saddle-point analysis at critical points of g to obtain a general asymptotic formula in terms of the critical set Psi with non-degenerate Hessians. As an application, it computes the average number E_k^c(n) of proper c-edge-colorings for random k-regular graphs, providing explicit leading terms for the cases c = k and c > k. The results generalize previous bicolored work, connect to Polya enumeration through automorphism weights, and offer a versatile method for asymptotic counting in colored regular graphs with potential for broader combinatorial applications.

Abstract

We prove a formula for the asymptotic number of edge-colored regular graphs with a prescribed set of allowed vertex-incidence structures. The formula depends on specific critical points of a polynomial encoding the vertex-incidences. As an application, we compute the expected number of proper $c$-edge-colorings of a large random $k$-regular graph.

Figures (2)

• Figure 1: An edge-colored $5$-regular graph, element of $\mathcal{G}_5^3(6)$.
• Figure :

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Lemma 3.1
• Remark 3.2
• proof
• Proposition 3.3