A study on T-equivalent graphs

Abstract

In his article [J. Comb. Theory Ser. B 16 (1974), 168–174], Tutte called two graphs T-equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs G and $G^{\prime }$ are T-equivalent if $G^{\prime }$ is obtained from G by flipping a rotor (i.e., replacing it by its mirror) of order at most 5, where a rotor of order k in G is an induced subgraph R having an automorphism ψ with a vertex orbit $\left\{{\psi }^i\right(u):i\ge 0\}$ of size k such that every vertex of R is only adjacent to vertices in R unless it is in this vertex orbit. In this article, we show the above result due to Tutte can be extended to a rotor R of order $k\ge 6$ if the subgraph of G induced by all those edges of G which are not in R satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic T-equivalent pairs of graphs.