Properties of Catlin’s reduced graphs and supereulerian graphs

A graph G is called collapsible if for every even subset R ⊆ V (G), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is the reduction of G if it is obtained from G by contracting all the nontrivial collapsible subgraphs. A graph is reduced if it has no nontrivial collapsible subgraphs. In this paper, we first prove a few results on the properties of reduced graphs. As an application, for 3-edge-connected graphs G of order n with d(u) + d(v) ≥ 2(n/p − 1) for any uv ∈ E(G) where p > 0 are given, we show how such graphs change if they have no spanning Eulerian subgraphs when p is increased from p = 1 to 10 then to 15.