On [formula omitted]-supereulerian graphs with linear degree bounds

Abstract

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<a<1, there exists a family of finitely many graphs F(a,b;s,t) such that if G is a simple graph on n vertices with κ′(G)≥t+2 and δ(G)≥an+b, then either G is (s,t)-supereulerian, or G is contractible to a member in F(a,b;s,t).(ii) Let ℓK2 denote the connected loopless graph with two vertices and ℓ parallel edges. If G is a simple graph on n vertices with κ′(G)≥t+2 and δ(G)≥n2−1, then when n is sufficiently large, either G is (s,t)-supereulerian, or for some integer j with t+2≤j≤s+t, G is contractible to a jK2.