The Lovasz-Cherkassky theorem for locally finite graphs with ends

Lovasz and Cherkassky discovered independently that, if G is a finite graph and T & SUBE; V(G) such that the degree dG(v) is even for every vertex v E V(G) .T, then the maximum number of edge-disjoint paths which are internally disjoint from T and connect distinct E vertices of T is equal to 1tET & lambda;G(t, T . {t}) (where & lambda;G(t, T . {t}) is the size of a smallest 2 cut that separates t and T {t}). From another perspective, this means that for every vertex t E T, in any optimal path-system there are & lambda;G(t, T {t}) many paths between t and T {t}. We extend the theorem of Lovasz 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.& COPY; 2023 Elsevier B.V. All rights reserved.