Spanning Trails in a 2Connected Graph
Abstract
In this article we prove the following: Let $G$ be a $2$connected graph with circumference $c(G)$. If $c(G)\leq 5$, then $G$ has a spanning trail starting from any vertex, if $c(G)\leq 7$, then $G$ has a spanning trail.
As applications of this result, we obtain the following.

Every $2$edgeconnected graph of order at most 8 has a spanning trail starting from any vertex with the exception of six graphs.

Let $G$ be a $2$edgeconnected graph and $S$ a subset of $V(G)$ such that $E(GS)=\emptyset$ and $S\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $S\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$.
 Every $2$connected clawfree graph $G$ with order $n$ and minimum degree $\delta(G)> \frac{n}{7}+4\geq 23$ is traceable or belongs to two exceptional families of welldefined graphs, and moreover, if $\delta(G)> \frac{n}{6}+4\geq 13$, then $G$ is traceable.
All above results are sharp in a sense.