### Abstract

Let G be a graph and W a subset of V(G). Let g, f : V(G) → Z be two integer-valued functions such that g(x) ≤ f(x) for all x ∈ V(G) and g(y) = f(y) (mod 2) for all y ∈ W. Then a spanning subgraph F of G is called a partial parity (g, f)-factor with respect to W if g(x) ≤ deg_{F}(x) ≤ f(x) for all x ∈ V(G) and deg_{F}(y) ≡ f(y) (mod 2) for all y ∈ W. We obtain a criterion for a graph G to have a partial parity (g, f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.

Original language | English |
---|---|

Pages (from-to) | 501-509 |

Number of pages | 9 |

Journal | Graphs and Combinatorics |

Volume | 17 |

Issue number | 3 |

Publication status | Published - 2001 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Graphs and Combinatorics*,

*17*(3), 501-509.

**Partial parity (g, f)-factors and subgraphs covering given vertex subsets.** / Kano, M.; Matsuda, Haruhide.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 17, no. 3, pp. 501-509.

}

TY - JOUR

T1 - Partial parity (g, f)-factors and subgraphs covering given vertex subsets

AU - Kano, M.

AU - Matsuda, Haruhide

PY - 2001

Y1 - 2001

N2 - Let G be a graph and W a subset of V(G). Let g, f : V(G) → Z be two integer-valued functions such that g(x) ≤ f(x) for all x ∈ V(G) and g(y) = f(y) (mod 2) for all y ∈ W. Then a spanning subgraph F of G is called a partial parity (g, f)-factor with respect to W if g(x) ≤ degF(x) ≤ f(x) for all x ∈ V(G) and degF(y) ≡ f(y) (mod 2) for all y ∈ W. We obtain a criterion for a graph G to have a partial parity (g, f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.

AB - Let G be a graph and W a subset of V(G). Let g, f : V(G) → Z be two integer-valued functions such that g(x) ≤ f(x) for all x ∈ V(G) and g(y) = f(y) (mod 2) for all y ∈ W. Then a spanning subgraph F of G is called a partial parity (g, f)-factor with respect to W if g(x) ≤ degF(x) ≤ f(x) for all x ∈ V(G) and degF(y) ≡ f(y) (mod 2) for all y ∈ W. We obtain a criterion for a graph G to have a partial parity (g, f)-factor with respect to W. Furthermore, by making use of this criterion, we give some necessary and sufficient conditions for a graph G to have a subgraph which covers W and has a certain given property.

UR - http://www.scopus.com/inward/record.url?scp=19544374330&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=19544374330&partnerID=8YFLogxK

M3 - Article

VL - 17

SP - 501

EP - 509

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 3

ER -