The Lovász-Cherkassky theorem for locally finite graphs with ends

Abstract

Lovász and Cherkassky discovered independently that, if G is a finite graph and T⊆V(G) such that the degree dG(v) is even for every vertex v∈V(G)∖T, then the maximum number of edge-disjoint paths which are internally disjoint from T and connect distinct vertices of T is equal to 12∑t∈TλG(t,T∖{t}) (where λG(t,T∖{t}) is the size of a smallest cut that separates t and T∖{t}). From another perspective, this means that for every vertex t∈T, in any optimal path-system there are λG(t,T∖{t}) many paths between t and T∖{t}. We extend the theorem of Lovász 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.