Supereulerian graphs with width s and s-collapsible graphs

For an integer s > 0 and for u, v ∈ V (G) with u ≠ v, an (s; u, v)-trail-system of G is a subgraphH consisting of s edge-disjoint (u, v)-trails. A graph is supereulerianwithwidth s if for any u, v ∈ V (G) with u ≠ v, G has a spanning (s; u, v)-trail-system. The supereulerian width μ(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer kwith 0 ≤ k ≤ s. Thus a graph Gwith μ(G) ≥ 2 has a spanning Eulerian subgraph. Catlin (1988) introduced collapsible graphs to study graphs with spanning Eulerian subgraphs, and showed that every collapsible graph G satisfies μ(G) ≥ 2 (Catlin, 1988; Lai et al., 2009). Graphs Gwithμ(G) ≥ 2 have also been investigated by Luo et al. (2006) as Eulerian-connected graphs. In this paper, we extend collapsible graphs to scollapsible graphs and develop a new related reduction method to study μ(G) for a graph G. In particular, we prove that K3,3 is the smallest 3-edge-connected graph with μ < 3. These results and the reductionmethodwill be applied to determine a best possible degree condition for graphs with supereulerian width at least 3, which extends former results in Catlin (1988) and Lai (1988). © 2015 Elsevier B.V. All rights reserved.