AI 中文总结
该文刻画了有限路宽可数图的特征，并证明线宽至多为$k$的图类存在线宽为$\boldsymbol{O(k^2)}$的普适图，同时给出有限路宽图的普适性相关结论。
AI 中文摘要
我们研究可数无限图中的路宽及与其密切相关的参数线宽。首个结果对有限路宽的图进行特征刻画：这类图不存在无限多个无限度顶点，不存在无限多个两两不交的无限路径，且不含最大度为3的有限树的细分图。随后，我们研究子图关系下路宽或线宽有界图的普适性，特别证明存在线宽为$\boldsymbol{O(k^2)}$的普适图，对应线宽至多为$k$的图类；与之相对，有限路宽的图无法成为路宽为1的局部有限图类的普适图；最后，对每个$k\boldsymbol{\times}2$，所有路宽至多为$k$的图类的普适图，其线宽至少为$k+1$。
英文摘要
We study path-width and the closely related parameter line-width in countably infinite graphs. Our first result characterises the graphs of finite path-width: they are the graphs that do not have infinitely many vertices of infinite degree, do not have infinitely many pairwise disjoint infinite paths, and contain no subdivision of some finite tree of maximum degree 3. We then investigate universality under the subgraph relation for graphs of bounded path-width or line-width. In particular, we prove that there exists a universal graph with line-width $\mathcal{O}(k^2)$ for the class of graphs with line-width at most $k$. In contrast, we show that no graph of finite path-width is universal for the class of locally finite graphs with path-width $1$. Finally, we show that for each $k\geq 2$, every universal graph for the class of graphs with path-width at most $k$ has line-width at least $k + 1$.
