Tutte polynomials and a stronger version of the Akiyama-Harary problem

TL;DR

This paper investigates the relationship between graphs and their complements through Tutte polynomials, disproving a conjecture about degree sequences and demonstrating infinitely many counterexamples.

Contribution

It provides a counterexample to a conjecture relating degree sequences of graphs and their complements, and extends the analysis to Tutte polynomials showing infinitely many such graphs exist.

Findings

Counterexamples to the conjecture about degree sequences.

Existence of infinitely many graphs with identical Tutte polynomials as their complements.

Abstract

Can a non self-complementary graph have the same chromatic polynomial as its complement? The answer to this question of Akiyama and Harrary is positive and was given by J. Xu and Z. Liu. They conjectured that every such graph has the same degree sequence as its complement. In this paper we show that there are infinitely many graphs for which this conjecture does not hold. We then solve a more general variant of the Akiyama-Harary problem by showing that there exists infinitely many non self-complementary graphs having the same Tutte polynomial as their complements.