交错多项式的不可约性
Irreducibility of interlace polynomials
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中文总结 AI 辅助
该研究证明二元交错多项式$q(G;x,y)$与Courcelle多元交错多项式$C_G(u,v;\boldsymbol{x},\boldsymbol{y})$的不可约性均等价于对应图$G$连通，且构造了反例说明二元交错多项式的无环假设是必要的。
中文摘要 AI 辅助
图多项式的因式分解通常反映组合分解。对于非空无环图$G$，我们首先证明，由Arratia、Bollobás和Sorkin提出的二元交错多项式$q(G;x,y)$在$\boldsymbol{\text{C}}[x,y]$上不可约当且仅当$G$是连通的，这与Tutte多项式的经典不可约性定理完全对应。无环假设是必要的：我们构造了一个无限族的连通带环图，其二元交错多项式是可约的。对于非空图$G$，我们证明Courcelle的多元交错多项式$C_G(u,v;\boldsymbol{x},\boldsymbol{y})$在$\boldsymbol{\text{C}}[u,v,x_a,y_a:a\in V(G)]$上不可约当且仅当$G$是连通的。
英文摘要
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.
