\((1,f)\) -Factors of Graphs with Odd Property

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• 作者：Yoshimi Egawa ; Mikio Kano ; Zheng Yan
• 关键词：Factor of graph ; $$( 1 ; f)$$ ( 1 ; f ) ; Odd factor ; Odd components
• 刊名：Graphs and Combinatorics
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：32
• 期：1
• 页码：103-110
• 全文大小：391 KB
• 参考文献：1.Akiyama, J., Kano, M.: Factors and Factorizations of Graphs LNM 1031. Springer, Berlin (2011)CrossRef
  2.Cui, Y., Kano, M.: Some results on odd factors of graphs. J. Graph Theory 12, 327–333 (1988)MATH MathSciNet CrossRef
  3.Lu, H.L., Wang, D.G.L.: On Cui–Kano’s characterization problem on graph factor. J. Graph Theory 74, 335–343 (2013)MATH MathSciNet CrossRef
  
• 作者单位：Yoshimi Egawa (1)
  Mikio Kano (2)
  Zheng Yan (3)
  
  1. Tokyo University of Science, Shinjuku-Ku, Tokyo, Japan
  2. Ibaraki University, Hitachi, Ibaraki, Japan
  3. Yangtzeu University, Jing Zhou, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Engineering Design
  
• 出版者：Springer Japan
• ISSN：1435-5914

文摘

Let \(G\) be a graph and \(f:V(G)\rightarrow \{1,2,3,4,\ldots \}\) be a function. We denote by \(odd(G)\) the number of odd components of \(G\). We prove that if \(odd(G-X)\le \sum _{x\in X}f(x)\) for all \( X\subset V(G)\), then \(G\) has a \((1,f)\)-factor \(F\) such that, for every vertex \(v\) of \(G\), if \(f(v)\) is even, then \(\deg _F(v)\in \{1,3,\ldots ,f(v)-1,f(v)\}\), and otherwise \(\deg _F(v)\in \{1,3, \ldots , f(v)\}\). This theorem is a generalization of both the \((1,f)\)-odd factor theorem and a recent result on \(\{1,3, \ldots , 2n-1,2n\}\)-factors by Lu and Wang. We actually prove a result stronger than the above theorem. Keywords Factor of graph \(( 1, f)\)-Odd factor Odd components