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 $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex with the exception of six graphs. Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\emptyset$ 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 claw-free 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 well-defined graphs, and moreover, if $\delta(G)> \frac{n}{6}+4\geq 13$, then $G$ is traceable. All above results are sharp in a sense.
@article{10_37236_8121,
author = {Shipeng Wang and Liming Xiong},
title = {Spanning trails in a 2-connected graph},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {3},
doi = {10.37236/8121},
zbl = {1420.05089},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8121/}
}
TY - JOUR
AU - Shipeng Wang
AU - Liming Xiong
TI - Spanning trails in a 2-connected graph
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8121/
DO - 10.37236/8121
ID - 10_37236_8121
ER -
%0 Journal Article
%A Shipeng Wang
%A Liming Xiong
%T Spanning trails in a 2-connected graph
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8121/
%R 10.37236/8121
%F 10_37236_8121
Shipeng Wang; Liming Xiong. Spanning trails in a 2-connected graph. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/8121
