Excluding long paths

TL;DR

This paper generalizes Ding's 1992 result on finite graphs to infinite graphs, including those with parallel edges and loops, establishing conditions for isomorphic induced subgraphs.

Contribution

It extends Ding's theorem from finite to infinite graphs, allowing for loops and parallel edges, broadening the scope of the original result.

Findings

Generalization of Ding's theorem to infinite graphs

Inclusion of graphs with loops and parallel edges

Existence of isomorphic induced subgraphs in infinite graph sequences

Abstract

Ding (1992) proved that for each integer ${m} \geqslant 0$, and every infinite sequence of finite simple graphs $G_1, G_2, \ldots$, if none of these graphs contains a path of length ${m}$ as a subgraph, then there are indices $i < j$ such that $G_i$ is isomorphic to an induced subgraph of $G_j$. We generalise this result to infinite graphs, possibly with parallel edges and loops.