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, then G 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 K 2 or to K 2 , l with l odd.