Geodesic
Two operations on a graph preserving the (non)existence of 2-factors in its line graph
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1035-1044.

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Let $G=(V(G),E(G))$ be a graph. Gould and Hynds (1999) showed a well-known characterization of $G$ by its line graph $L(G)$ that has a 2-factor. In this paper, by defining two operations, we present a characterization for a graph $G$ to have a 2-factor in its line graph $L(G).$ A graph $G$ is called $N^{2}$-locally connected if for every vertex $x\in V(G),$ $G[\{y\in V(G)\; 1\leq {\rm dist}_{G}(x,y)\leq 2\}]$ is connected. By applying the new characterization, we prove that every claw-free graph in which every edge lies on a cycle of length at most five and in which every vertex of degree two that lies on a triangle has two $N^{2}$-locally connected adjacent neighbors, has a $2$-factor. This result generalizes the previous results in papers: Li, Liu (1995) and Tian, Xiong, Niu (2012), and is the best possible.
DOI : 10.1007/s10587-014-0151-4
Classification : 05C35, 05C38, 05C45
Keywords: 2-factor; claw-free graph; line graph; $N^{2}$-locally connected 
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     title = {Two operations on a graph preserving the (non)existence of 2-factors in its line graph},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1035--1044},
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An, Mingqiang; Lai, Hong-Jian; Li, Hao; Su, Guifu; Tian, Runli; Xiong, Liming. Two operations on a graph preserving the (non)existence of 2-factors in its line graph. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 1035-1044. doi : 10.1007/s10587-014-0151-4. http://geodesic.mathdoc.fr/articles/10.1007/s10587-014-0151-4/

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