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A study on $T$-equivalent graphs

Fengming Dong, Meiqiao Zhang

TL;DR

The paper investigates graphs with identical Tutte polynomials (T-equivalence) and expands Tutte's rotor-flip result from rotor order at most $5$ to orders $k\ge6$ under specific conditions on the non-rotor subgraph. It establishes necessary and sufficient criteria for when combining rotor endpoints with an arbitrary graph $W$ preserves $T$-equivalence, and introduces a partition-based viewpoint that strengthens the theoretical framework. A key contribution is a constructive method to generate infinitely many new $T$-equivalent pairs by exploiting a digraph $D_{\psi}$ derived from an isomorphism between contracted graphs, attaching gadget graphs along directed cycles, thereby producing rich families of non-isomorphic examples. The results deepen the understanding of how the Tutte polynomial encodes graph structure and provide practical tools for building large families of $T$-equivalent graphs with prescribed rotor configurations.

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 $G'$ are $T$-equivalent if $G'$ 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 $\{ψ^i(u): i\ge 0\}$ 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 first 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.

A study on $T$-equivalent graphs

TL;DR

The paper investigates graphs with identical Tutte polynomials (T-equivalence) and expands Tutte's rotor-flip result from rotor order at most to orders under specific conditions on the non-rotor subgraph. It establishes necessary and sufficient criteria for when combining rotor endpoints with an arbitrary graph preserves -equivalence, and introduces a partition-based viewpoint that strengthens the theoretical framework. A key contribution is a constructive method to generate infinitely many new -equivalent pairs by exploiting a digraph derived from an isomorphism between contracted graphs, attaching gadget graphs along directed cycles, thereby producing rich families of non-isomorphic examples. The results deepen the understanding of how the Tutte polynomial encodes graph structure and provide practical tools for building large families of -equivalent graphs with prescribed rotor configurations.

Abstract

In his article [J. Comb. Theory Ser. B 16 (1974), 168-174], Tutte called two graphs -equivalent (i.e., codichromatic) if they have the same Tutte polynomial and showed that graphs and are -equivalent if is obtained from by flipping a rotor (i.e., replacing it by its mirror) of order at most , where a rotor of order in is an induced subgraph having an automorphism with a vertex orbit of size such that every vertex of is only adjacent to vertices in unless it is in this vertex orbit. In this article, we first show the above result due to Tutte can be extended to a rotor of order if the subgraph of induced by all those edges of which are not in satisfies certain conditions. Also, we provide a new method for generating infinitely many non-isomorphic -equivalent pairs of graphs.
Paper Structure (5 sections, 11 theorems, 19 equations, 4 figures)

This paper contains 5 sections, 11 theorems, 19 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Necessary and sufficient conditions
  4. Extension of Theorem \ref{['theo-tutte']}
  5. Generating $T$-equivalent pairs

Key Result

Theorem 1

Let $R$ and $W$ be connected graphs. If $w_1,\dots,w_k$ are distinct vertices in $W$, and $\{u_i: i\in [k]\}$ is a vertex orbit of an automorphism $\psi$ in $R$ such that $\psi(u_k)=u_{1}$ and $\psi(u_i)=u_{i+1}$ for all $i\in [k-1]$, where $k\in [5]$, then $R(u_{1},\dots,u_k)\sqcup W(w_1,\dots,w_k)

Figures (4)

  • Figure 1: $G'$ is obtained from $G$ by a Whitney twist
  • Figure 2: Graphs $G,W$ and $G(u_1,u_2,u_3)\sqcup W(w_1,w_2,w_3)$
  • Figure 3: Graphs $G^*$ and $H^*$
  • Figure 4: Examples for graphs $W_0$ with $r=3$ and $g=2$

Theorems & Definitions (11)

  • Theorem 1: Tutte tutte
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Theorem 2
  • Proposition 3
  • Theorem 3
  • Proposition 4
  • Theorem 4
  • Proposition 5
  • ...and 1 more