Saddle-Point Asymptotics for Chromatic and Tutte Polynomial Evaluations of Complete Multipartite Graphs

TL;DR

This paper develops a saddle-point theory for evaluating chromatic and Tutte polynomials of complete multipartite graphs, providing asymptotic formulas and confirming conjectures with applications to combinatorial sequences.

Contribution

It introduces a Gamma-type integral representation for acyclic orientations, proves fixed-column conjectures, and develops a multivariable analytic framework for graph polynomial evaluations.

Findings

Proves fixed-column conjecture for A267383 for any fixed number of parts.

Derives fixed-p Tutte-axis asymptotics for complete multipartite graphs.

Establishes logarithmic asymptotics for specific partition-sum sequences.

Abstract

We develop a saddle-point theory for acyclic orientations and negative chromatic evaluations of complete multipartite graphs, with applications to OEIS A267383, A372326, A372084, A372395, and A370613. The main tool is an exact Gamma-type integral representation for acyclic orientation counts and its Gamma-weighted extension to the negative chromatic axis. We prove Kotesovec's fixed-column conjecture for A267383 for arbitrary fixed numbers of parts, give the corresponding fixed-p Tutte-axis asymptotics, develop an analytic-combinatorics-in-several-variables framework for chromatic evaluations of fixed graph blow-ups, and give unconditional fixed-base families reducible to balanced Turan graphs. In the product regimes we prove fixed part-size and finite-profile expansions, and for equal-size parts we obtain an all-order expansion throughout every fixed polynomial window, including explicit corrections through the cubic scale. Finally, we prove logarithmic asymptotics for the partition-sum sequences A372395 and A370613 via a quadratic-energy partition model, a growing-window comparison for the Stirling-transform factors, and a random-permutation far-tail bound.