Supereulerian graphs with width s and s-collapsible graphs

摘要

For an integers > 0 and for u, v is an element of V(G) with u not equal v, an (s; u, v)-trail-system of G is a subgraph H consisting of s edge-disjoint (u, v)-trails. A graph is supereulerian with width s if for any u, v is an element of V(G) with u not equal v, G has a spanning (s; u, v)-trail-system. The supereulerian width mu'(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer k with 0 <= k <= s. Thus a graph G with mu'(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 mu'(G) >= 2 (Catlin, 1988; Lai et al., 2009). Graphs G with mu'(G) >= 2 have also been investigated by Luo et al. (2006) as Eulerian-connected graphs. In this paper, we extend collapsible graphs to s collapsible graphs and develop a new related reduction method to study mu'(G) for a graph G. In particular, we prove that K-3,K-3 is the smallest 3-edge-connected graph with mu' < 3. These results and the reduction method will 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). (C) 2015 Elsevier B.V. All rights reserved.