Packing Euler graphs with traces

Abstract

For a graph G = (V,E) and a vertex v ∈ V, let T(v) be a local trace at v, i.e. T(v) is an Eulerian subgraph of G such that every walkW(v), with start vertex v can be extended to an Eulerian tour in T(v). In general, local traces are not unique. We prove that if G is Eulerian every maximum edge-disjoint cycle packing Z* of G induces maximum local traces T(v) at v for every v ∈ V. In the opposite, if the total size $$ \sum $$V∈E|(T(v)|| is minimal then the set of related edge-disjoint cycles in G must be maximum.