### Abstract

A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. Also, M is said to be a distance d matching if the shortest distance between a pair of edges in M is at least d. A graph G is distance d matchable if every distance d matching is extendable in G, regardless of its size. In this paper, we study the class of distance d matchable graphs. In particular, we prove that for every integer K with k ≥ 3, there exists a positive integer d such that every connected, locally (k − 1)-connected K_{1,k}-free graph of even order is distance d matchable. We also prove that every connected, locally K-connected K_{1,f}-free graph of even order is distance 3 matchable. Furthermore, we make more detailed analysis of K_{1,4}-free graphs and study their distance matching extension properties.

Original language | English |
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Journal | Journal of Graph Theory |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### Keywords

- distance restricted matching extension
- local connectivity
- star-free

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22465

**Distance matching extension and local structure of graphs.** / Aldred, R. E.L.; Fujisawa, Jun; Saito, Akira.

Research output: Contribution to journal › Article

*Journal of Graph Theory*. https://doi.org/10.1002/jgt.22465

}

TY - JOUR

T1 - Distance matching extension and local structure of graphs

AU - Aldred, R. E.L.

AU - Fujisawa, Jun

AU - Saito, Akira

PY - 2019/1/1

Y1 - 2019/1/1

N2 - A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. Also, M is said to be a distance d matching if the shortest distance between a pair of edges in M is at least d. A graph G is distance d matchable if every distance d matching is extendable in G, regardless of its size. In this paper, we study the class of distance d matchable graphs. In particular, we prove that for every integer K with k ≥ 3, there exists a positive integer d such that every connected, locally (k − 1)-connected K1,k-free graph of even order is distance d matchable. We also prove that every connected, locally K-connected K1,f-free graph of even order is distance 3 matchable. Furthermore, we make more detailed analysis of K1,4-free graphs and study their distance matching extension properties.

AB - A matching M in a graph G is said to be extendable if there exists a perfect matching of G containing M. Also, M is said to be a distance d matching if the shortest distance between a pair of edges in M is at least d. A graph G is distance d matchable if every distance d matching is extendable in G, regardless of its size. In this paper, we study the class of distance d matchable graphs. In particular, we prove that for every integer K with k ≥ 3, there exists a positive integer d such that every connected, locally (k − 1)-connected K1,k-free graph of even order is distance d matchable. We also prove that every connected, locally K-connected K1,f-free graph of even order is distance 3 matchable. Furthermore, we make more detailed analysis of K1,4-free graphs and study their distance matching extension properties.

KW - distance restricted matching extension

KW - local connectivity

KW - star-free

UR - http://www.scopus.com/inward/record.url?scp=85066493458&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066493458&partnerID=8YFLogxK

U2 - 10.1002/jgt.22465

DO - 10.1002/jgt.22465

M3 - Article

AN - SCOPUS:85066493458

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

ER -