On a conjecture of Las Vergnas

Abstract

In 1988, Las Vergnas conjectured that if $M$ is a binary matroid with bicycle dimension $d$, then for $0 \leq k \leq d$, the $k$th derivative of the diagonal Tutte polynomial $T(M;z,z)$ evaluated at $z=-1$ is an integer multiple of $2^{d-k}$. While this was rapidly disproved for binary matroids and for graphs in general, extensive computations strongly suggested that it might be true for planar graphs. In this paper we prove that this is indeed the case. To do this, we consider a stronger divisibility property that we call the LV property, and a larger class of graphs, namely the class of delta-wye-reducible graphs. By a detailed analysis of how a delta-wye exchange affects the coefficients of the diagonal Tutte polynomial, we show that delta-wye-reducible graphs have the LV property. That Las Vergnas' conjecture holds for planar graphs immediately follows because planar graphs are delta-wye reducible and the LV property is stronger than Las Vergnas' divisibility conditions.