Geodesic
Even factor of bridgeless graphs containing two specified edges
Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1105-1114
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An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$ in which each component containing neither $e_1$ nor $e_2$ has order at least four. In this paper we improve this result and prove that $G$ has an even factor containing $e_1$ and $e_2$ such that each component has order at least four.
An even factor of a graph is a spanning subgraph in which each vertex has a positive even degree. Let $G$ be a bridgeless simple graph with minimum degree at least $3$. Jackson and Yoshimoto (2007) showed that $G$ has an even factor containing two arbitrary prescribed edges. They also proved that $G$ has an even factor in which each component has order at least four. Moreover, Xiong, Lu and Han (2009) showed that for each pair of edges $e_1$ and $e_2$ of $G$, there is an even factor containing $e_1$ and $e_2$ in which each component containing neither $e_1$ nor $e_2$ has order at least four. In this paper we improve this result and prove that $G$ has an even factor containing $e_1$ and $e_2$ such that each component has order at least four.
DOI : 10.21136/CMJ.2018.0114-17
Classification : 05C70
Keywords: bridgeless graph; components of an even factor; specified edge 
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Haghparast, Nastaran; Kiani, Dariush. Even factor of bridgeless graphs containing two specified edges. Czechoslovak Mathematical Journal, Tome 68 (2018) no. 4, pp. 1105-1114. doi: 10.21136/CMJ.2018.0114-17

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[5] Xiong, L., Lu, M., Han, L.: The structure of even factors in claw-free graphs. Discrete Math. 309 (2009), 2417-2423. | DOI | MR | JFM

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