Countable Graphs with Finite Path-width: Characterisation and Universality

2026-08-27

AI Summary

研究了可数无限图的路径宽度和线宽，通过特定条件刻画有限路径宽度的图，并探讨了具有有界路径或线宽图的普遍性问题。

Abstract

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 $k + 1$.

Problem

• path-width
• line-width
• countably infinite graphs
• universality
• subgraph relation

Innovation

• path-width
• line-width
• universality
• countably infinite graphs