Abstract
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.
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M. Kano was supported by JSPS KAKENHI Grant Number 25400187. Z. Yan is a joint corresponding author.
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Egawa, Y., Kano, M. & Yan, Z. \((1,f)\)-Factors of Graphs with Odd Property. Graphs and Combinatorics 32, 103–110 (2016). https://doi.org/10.1007/s00373-015-1558-x
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DOI: https://doi.org/10.1007/s00373-015-1558-x