A recurrence for certain Tutte polynomials
Vincent Brugidou
TL;DR
This work presents a new two-variable recurrence for the Tutte polynomials $T_n^{(r)}(x,y)$ of graphs obtained by contracting the complete graph $K_n$ along $r$ vertices, generalizing a prior one-variable relation involving the inversion enumerator polynomials $J_n^{(r)}(q)$. The authors provide a combinatorial proof based on the Tutte polynomial expansion, decomposing spanning subgraphs by the neighbor set $S$ of the contracted vertex and analyzing cases $|S|=0$, $|S|=1$, and $|S|\ge2$ via contraction maps to derive the recurrence $T_n^{(r)}(x,y)=\sum_{s=1}^{n-r} {n-r \choose s} [r]_y^{s} y^{\binom{s}{2}} T_{n-r}^{(s)}(x,y) + (x-1) T_{n-r}^{(1)}(x,y)$. The paper also shows how this specializes to connected graphs, yielding $C_n^{(r)}(t)=t^{n-r} J_n^{(r)}(1+t)$ and recovering the known $r=1$ case, with a triangular array of polynomials and practical recursion for computing $T_n^{(r)}$. Overall, the results connect Tutte polynomials of contracted complete graphs to inversion enumerators and parking-function frameworks, providing new combinatorial tools for these polynomials.
Abstract
We combinatorially prove a new recurrence between the Tutte polynomials of graphs obtained by contraction of the complete graphs $K_{n}$%. This generalizes, to two variables, a relation previously obtained by the author between the inversion enumerator polynomials in the colored tree sequences.