Asymptotic number of edge-colored regular graphs

TL;DR

This paper derives an asymptotic formula for counting edge-colored regular graphs with specified vertex-incidence patterns and applies it to estimate the expected number of proper edge-colorings in large random regular graphs.

Contribution

It introduces a new asymptotic counting formula for edge-colored regular graphs based on critical points of a polynomial, advancing combinatorial enumeration methods.

Findings

Derived an explicit asymptotic formula for edge-colored regular graphs.

Computed the expected number of proper edge-colorings in large random regular graphs.

Connected combinatorial enumeration with probabilistic properties of graph colorings.

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.