Eulerian subgraphs containing given edges

Abstract

For an integer l⩾0, define SE(l) to be the family of graphs such that G∈ SE(l) if and only if for any edge subset X⊆ E( G) with | X|⩽ l, G has a spanning eulerian subgraph H with X⊆ E( H). The graphs in SE(0) are known as supereulerian graphs. Let f( l) be the minimum value of k such that every k-edge-connected graph is in SE(l) . Jaeger and Catlin independently proved f(0)=4. We shall determine f( l) for all values of l⩾0. Another problem concerning the existence of eulerian subgraphs containing given edges is also discussed, and former results in [J. Graph Theory 1 (1977) 79–84] and [J. Graph Theory 3 (1979) 91–93] are extended.