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  F 1  and   F 2  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  F 1 .

(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  F 2 .