Strengthened Ore conditions for (s,t)-supereulerian graphs

For integers and , a graph is -supereulerian if for any disjoint edge sets with and , has a spanning closed trail that contains and avoids . Pulleyblank (1979) showed that determining whether a graph is -supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in (Catlin, 1988) showed that every simple graph on vertices with , when is sufficiently large, is -supereulerian or is contractible to . A function has been found that every -supereulerian graph must have edge connectivity at least .For any nonnegative integers and , we obtain best possible Ore conditions to assure a simple graph on vertices to be -supereulerian as stated in the following.(i) For any real numbers and with , there exists a family of finitely many graphs such that if and if for any nonadjacent vertices , , then either is -supereulerian, or is contractible to a member in .(ii) If and if for any nonadjacent vertices , , then when is sufficiently large, either is -supereulerian, or is contractible to one of the six well specified graphs.(iii) Suppose that . If then is -supereulerian if and only if .