Smallest k-edge-connected claw-free graphs without special spanning trails

Harris and Mossinghoff [7] and Bullock, Frick and Singleton [3] determined the smallest 2-connected non-traceable claw-free graphs G₁, G ₂ . In this paper, we consider the analogues for 2-edge-connected graphs and trails, determine the smallest 2-edge-connected claw-free graphs without a spanning trail and the smallest 2-edge-connected non-traceable claw-free graphs. It is interesting that the graphs of the first conclusion are also G₁,G₂, and the second can be deduced by the line graph closure of Ryjáček and by the following result: if G has no cut-vertex of degree two and of order at most 10, then either G has a dominating trail, or G belongs to one of four special graphs. We also determine the smallest K-edge-connected non-supereulerian claw-free graph for k = 2,3.