2-factors in claw-free graphs with locally disconnected vertices

An, Mingqiang, Xiong, Liming, and Tian, Runli. "2-factors in claw-free graphs with locally disconnected vertices." Czechoslovak Mathematical Journal 65.2 (2015): 317-330. <http://eudml.org/doc/270132>.

@article{An2015,
abstract = {An edge of $G$ is singular if it does not lie on any triangle of $G$; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G[N(u)]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle of length at least $4$ with at most $s$ non-singular edges and with at least $s-5$ locally connected vertices; (ii) for each locally disconnected vertex $v$ of degree $2$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle $C$ with at most $s$ non-singular edges and with at least $s-3$ locally connected vertices and such that $G[V (C)\cap V_\{2\} (G)]$ is a path or a cycle, then $G$ has a 2-factor, and it is the best possible in some sense. This result generalizes two known results in Faudree, Faudree and Ryjáček (2008) and in Ryjáček, Xiong and Yoshimoto (2010).},
author = {An, Mingqiang, Xiong, Liming, Tian, Runli},
journal = {Czechoslovak Mathematical Journal},
keywords = {claw-free graph; 2-factor; closure; locally disconnected vertex; singular edge},
language = {eng},
number = {2},
pages = {317-330},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {2-factors in claw-free graphs with locally disconnected vertices},
url = {http://eudml.org/doc/270132},
volume = {65},
year = {2015},
}

TY - JOUR
AU - An, Mingqiang
AU - Xiong, Liming
AU - Tian, Runli
TI - 2-factors in claw-free graphs with locally disconnected vertices
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 317
EP - 330
AB - An edge of $G$ is singular if it does not lie on any triangle of $G$; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G[N(u)]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle of length at least $4$ with at most $s$ non-singular edges and with at least $s-5$ locally connected vertices; (ii) for each locally disconnected vertex $v$ of degree $2$ in $G,$ there is a nonnegative integer $s$ such that $v$ lies on an induced cycle $C$ with at most $s$ non-singular edges and with at least $s-3$ locally connected vertices and such that $G[V (C)\cap V_{2} (G)]$ is a path or a cycle, then $G$ has a 2-factor, and it is the best possible in some sense. This result generalizes two known results in Faudree, Faudree and Ryjáček (2008) and in Ryjáček, Xiong and Yoshimoto (2010).
LA - eng
KW - claw-free graph; 2-factor; closure; locally disconnected vertex; singular edge
UR - http://eudml.org/doc/270132
ER -