Spanning 2‐trails from degree sum conditions

Abstract

AbstractSuppose G is a simple connected n‐vertex graph. Let σ3(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G 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 σ3G)≥ 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 (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. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004