Spanning trails with variations of Chvátal-Erdős conditions

Let α(G), α(G), κ(G) and κ (G) denote the independence number, the matching number, connectivity and edge connectivity of a graph G, respectively. We determine the finite graph families F1 and F2 such that each of the following holds. (i) If a connected graph G satisfies κ (G) ≥ α(G) − 1, then G has a spanning closed trail if and only if G is not contractible to a member of F1. (ii) If κ (G) ≥ max{2, α(G)− 3}, then G has a spanning trail. This result is best possible. (iii) If a connected graph G satisfies κ (G) ≥ 3 and α(G) ≤ 7, then G has a spanning closed trail if and only if G is not contractible to a member of F2. © 2016 Elsevier B.V. All rights reserved.