Irreducibility of interlace polynomials

Abstract

The factorisation of graph polynomials often reflects combinatorial decomposition. For a nonempty loopless graph $G$, we first prove that the two-variable interlace polynomial $q(G;x,y)$, introduced by Arratia, Bollobás and Sorkin, is irreducible over $\mathbb{C}[x,y]$ if and only if $G$ is connected, exactly paralleling the classical irreducibility theorem for the Tutte polynomial. The loopless hypothesis is essential: we construct an infinite family of connected looped graphs whose two-variable interlace polynomials are reducible. For a nonempty graph $G$, we prove that Courcelle's multivariate interlace polynomial $C_G(u,v;\mathbf{x},\mathbf{y})$ is irreducible over $\mathbb{C}[u,v,x_a,y_a:a\in V(G)]$ if and only if $G$ is connected.