On (s, t)-supereulerian graphs in locally highly connected graphs

Given two nonnegative integers s and t , a graph G is (s, t)-supereulerian if for any disjoint sets X, Y ⊂ E(G) with |X | ≤ s and |Y | ≤ t , there is a spanning eulerian subgraph H of G that contains X and avoids Y . We prove that if G is connected and locally k-edge-connected, thenG is (s, t)-supereulerian, for any pair of nonnegative integers s and t with s+t ≤ k−1. We further show that if s + t ≤ k and G is a connected, locally k-edge-connected graph, then for any disjoint sets X, Y ⊂ E(G)with |X | ≤ s and |Y ≤ t , there is a spanning eulerian subgraph H that contains X and avoids Y , if and only if G− Y is not contractible to K2 or to K2,l with l odd. © 2009 Elsevier B.V. All rights reserved.