The Lov\'{a}sz-Cherkassky theorem for locally finite graphs with ends

TL;DR

This paper extends the Lovász-Cherkassky theorem from finite graphs to locally finite infinite graphs, incorporating ends and infinite paths, thus broadening its applicability in infinite graph theory.

Contribution

The paper generalizes the Lovász-Cherkassky theorem to infinite graphs with ends, allowing for infinite paths and a broader set of terminal points.

Findings

Extended the theorem to locally finite infinite graphs

Included ends as terminal points in the theorem

Allowed infinite paths for end connections

Abstract

Lov\'{a}sz and Cherkassky discovered independently that, if $G$ is a finite graph and $T\subseteq V(G)$ such that the degree $d_G(v)$ is even for every vertex $v\in V(G)\setminus T$, then the maximum number of edge-disjoint paths which are internally disjoint from~$T$ and connect distinct vertices of $T$ is equal to $\frac{1}{2} \sum_{t\in T}\lambda_G(t, T\setminus \{t\})$ (where $\lambda_G(t, T\setminus \{t\})$ is the size of a smallest cut that separates $t$ and $T\setminus\{t\}$). From another perspective, this means that for every vertex $t\in T$, in any optimal path-system there are $\lambda_G(t, T\setminus \{t\})$ many paths between $t$ and~$T\setminus\{t\}$. We extend the theorem of Lov\'{a}sz and Cherkassky based on this reformulation to all locally-finite infinite graphs and their ends. In our generalisation, $T$ may contain not just vertices but ends as well, and paths are one-way (two-way) infinite when they establish a vertex-end (end-end) connection.