Spanning trails with variations of Chvátal–Erdős conditions

Let α(G)α(G), α′(G)α′(G), κ(G)κ(G) and κ′(G)κ′(G) denote the independence number, the matching number, connectivity and edge connectivity of a graph GG, respectively. We determine the finite graph families F1F1 and F2F2 such that each of the following holds.(i) If a connected graph GG satisfies κ′(G)≥α(G)−1κ′(G)≥α(G)−1, then GG has a spanning closed trail if and only if GG is not contractible to a member of F1F1.(ii) If κ′(G)≥max{2,α(G)−3}κ′(G)≥max{2,α(G)−3}, then GG has a spanning trail. This result is best possible.(iii) If a connected graph GG satisfies κ′(G)≥3κ′(G)≥3 and α′(G)≤7α′(G)≤7, then GG has a spanning closed trail if and only if GG is not contractible to a member of F2F2.