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

Abstract

AbstractGiven 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 K2 or to K2,l with l odd