On (s,t)-supereulerian graphs with linear degree bounds

For integers s≥0 and t≥0, a graph G is (s,t)-supereulerian if for any disjoint edge sets X,Y⊆E(G) with |X|≤s and |Y|≤t, G has a spanning closed trail that contains X and avoids Y. Pulleyblank in [J. Graph Theory, 3 (1979) 309-310] showed that determining whether a graph is (0,0)-supereulerian, even when restricted to planar graphs, is NP-complete. Settling an open problem of Bauer, Catlin in [J. Graph Theory, 12 (1988) 29–45] showed that every simple graph G on n vertices with δ(G)≥n5−1, when n is sufficiently large, is (0,0)-supereulerian or is contractible to K2,3. We prove the following for any nonnegative integers s and t. (i) For any real numbers a and b with 0