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- Title
$$(1,f)$$ -Factors of Graphs with Odd Property.
- Authors
Egawa, Yoshimi; Kano, Mikio; Yan, Zheng
- 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.
- Subjects
FACTORS (Algebra); GRAPH theory; NUMBER theory; MATHEMATICAL proofs; GEOMETRIC vertices; MATHEMATICS theorems
- Publication
Graphs & Combinatorics, 2016, Vol 32, Issue 1, p103
- ISSN
0911-0119
- Publication type
Article
- DOI
10.1007/s00373-015-1558-x