色多项式与列表着色函数相等性的非持久性
Non-persistence of equality between chromatic polynomials and list-color functions
AI总结：
本文否定了“若图G的色多项式与列表着色函数在k处相等且为正，则在k+1处也相等”的问题，对每个k≥3构造了无限族反例，并将结论推广到任意此类图的衍生图上。
AI中文摘要：
对于任意图G，令P(G,k)和P_ℓ(G,k)分别表示G的色多项式和列表着色函数。一个未解决的问题是：对任意图G和整数k，若等式P(G,k)=P_ℓ(G,k)>0成立，是否必有P(G,k+1)=P_ℓ(G,k+1)也成立？本文否定了该问题的答案。对每个整数k≥3，我们构造了无限族图G，满足P(G,k)=P_ℓ(G,k)>0，但P(G,k+1)>P_ℓ(G,k+1)。此外，以该无限族图为附着装置，我们进一步证明：任何满足P(H,k)=P_ℓ(H,k)>0的图H，都可衍生出无限族图H'，使得P(H',k)=P_ℓ(H',k)>0且P(H',k+1)>P_ℓ(H',k+1)。
英文摘要：
For any graph $G$, let $P(G,k)$ and $P_{\ell}(G,k)$ denote the chromatic polynomial and the list-color function of $G$, respectively. It remains an open problem whether, for every graph $G$ and integer $k$, the equality $P(G,k)=P_{\ell}(G,k)>0$ implies that $P(G,k+1)=P_{\ell}(G,k+1)$ also holds. In this paper, we answer this question in the negative. For every integer $k\ge 3$, we construct an infinite family of graphs $G$ such that $P(G,k)=P_{\ell}(G,k)>0$ while $P(G,k+1)>P_{\ell}(G,k+1)$. Moreover, using this infinite family of graphs as attachment gadgets, we further show that any graph $H$ with $P(H,k)=P_{\ell}(H,k)>0$ can be developed into an infinite family of graphs $H'$ with $P(H',k)=P_{\ell}(H',k)>0$ and $P(H',k+1)>P_{\ell}(H',k+1)$.
