Spanning 2-trails from degree sum conditions

Suppose G is a simple connected n-vertex graph. Let σ3(G) denote the minimum degree sum of three independent vertices in G (which is ∞ ifG has no set of three independent vertices). A 2-trail is a trail that uses every vertex at most twice. Spanning 2-trails generalize hamilton paths and cycles. We prove three main results. First, if σ3(G) ≥ n−1, then G has a spanning 2-trail, unless G ∼= K1,3. Second, if σ3(G) ≥ n, then G has either a hamilton path or a closed spanning 2-trail. Third, if G is 2-edge-connected and σ3(G) ≥ n, then G has a closed spanning 2-trail, unless G ∼= K2,3 or K∗ 2,3 (the 6-vertex graph obtained from K2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2-edge-connected factors, spanning k-walks, even factors, and supereulerian graphs. In particular, a closed spanning 2-trail may be regarded as a connected (and 2-edge-connected) even [2, 4]-factor. ∗Supported by NSF Grant DMS-0070613 †Supported by NSF Grant DMS-0070430