AI 中文总结
该研究刻画了$\text{AO}$-哈密顿的多路径，提出保持$\text{AO}$-哈密顿性的粘合准则与方法，定义无环多项式并探究其性质，为$\text{AO}$-哈密顿性的归纳认证提供支撑。
AI 中文摘要
我们研究图G的无环定向构成的图$\boldsymbol{\text{AO}}(G)$，该图中两个无环定向若仅在一条弧的取向上不同则相邻。我们重点关注$\text{AO}(G)$的哈密顿性，通过两种推广Brenner等人的锯齿方法的模式编织方法，刻画了哪些多路径是$\text{AO}$-哈密顿的；还给出了在给定图上粘合多路径以保持$\text{AO}$-哈密顿性的准则。为通过2-连通图的开耳分解对$\text{AO}$-哈密顿性进行归纳认证，我们提出了三种在给定图上粘合多个多路径的方法。此外，我们定义了无环多项式，以同时包含图的无环定向数量及Savage、Squire和West提出的“奇偶性问题”：若G的无环多项式不以-1为根，则G不是$\text{AO}$-哈密顿的。我们探究了无环多项式的诸多性质，证明其并非著名的Tutte-Whitney多项式的实例，但也呈现出部分删除-收缩现象。
英文摘要
We study the graph $\mathcal{AO}(G)$ of acyclic orientations of a graph $G$. Two acyclic orientations are adjacent in this graph if they disagree on the orientation of a single arc. In particular, we focus on the Hamiltonicity of the graphs $\mathcal{AO}(G)$. Using two methods of pattern lacing which generalize the zig-zag method of Brenner, Cardinal, McConville, Merino and Mütze, we characterize which multipaths are $\mathcal{AO}$-Hamiltonian. Moreover, we give a criterion for the gluing of a multipath on a given graph to preserve $\mathcal{AO}$-Hamiltonicity. Building towards an inductive certification of $\mathcal{AO}$-Hamiltonicity via the (open) ear decomposition of 2-connected graphs, we propose three ways of gluing several multipaths to a given graph. In addition, we define the acyclic polynomials to encapsulate both the number of acyclic orientations of a graph and the "parity problem" proposed by Savage, Squire and West: if $-1$ is not a root of the acyclic polynomial of $G$, then $G$ is not $\mathcal{AO}$-Hamiltonian. We explore numerous properties of the acyclic polynomials, proving that they are not instances of the famous Tutte-Whitney polynomials, but that they too exhibit a partial deletion-contraction phenomenon.
Comments58 pages, 19 figures, 5 tables
