# Perfect matchings in highly cyclically connected regular graphs

@article{Lukoka2017PerfectMI, title={Perfect matchings in highly cyclically connected regular graphs}, author={Robert Lukoťka and Edita Rollov'a}, journal={Journal of Graph Theory}, year={2017}, volume={100}, pages={28 - 49} }

A leaf matching operation on a graph consists of removing a vertex of degree 1 together with its neighbour from the graph. Let G be a d ‐regular cyclically ( d − 1 + 2 k ) ‐edge‐connected graph of even order, where k ≥ 0 and d ≥ 3 . We prove that for any given set X of d − 1 + k edges, there is no 1‐factor of G avoiding X if and only if either an isolated vertex can be obtained by a series of leaf matching operations in G − X , or G − X has an independent set that contains more than half of the…

## References

SHOWING 1-10 OF 24 REFERENCES

### Lower bound of cyclic edge connectivity for n-extendability of regular graphs

- MathematicsDiscret. Math.
- 1993

### k-factors in regular graphs

- Mathematics
- 2008

Plesnik in 1972 proved that an (m − 1)-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m − 1 edges. Alder et al. in…

### Maximally matched and super matched regular graphs

- MathematicsInt. J. Comput. Math. Comput. Syst. Theory
- 2016

By using perfect matching polytope, a 0–1 integer linear programming for matching preclusion number of general graph is presented and simple characterizations for maximally matched and super matched regular graphs are obtained.

### On Moore Graphs with Diameters 2 and 3

- MathematicsIBM J. Res. Dev.
- 1960

The proof exploits the characteristic roots and vectors of the adjacency matrix (and its principal submatrices) of the graph to prove the existence of connected, undirected graphs homogeneous of degree d and of diameter k.

### A note on 1-factors in graphs

- Mathematics
- 1979

Conditions on a graphG are presented which are sufficient to guarantee thatG−Z contains a 1-factor, whereZ is a set of edges ofG of restricted cardinality. These conditions provide generalizations of…

### Matching preclusion and conditional matching preclusion for regular interconnection networks

- MathematicsDiscret. Appl. Math.
- 2012

### 2‐factors with prescribed and proscribed edges

- MathematicsJ. Graph Theory
- 2005

Let G be a finite k‐edge‐connected simple graph. We consider when a set of independent edges can be extended to a 2‐factor such that this 2‐factor avoids a fixed set of independent edges. A complete…

### Matching preclusion and conditional matching preclusion for bipartite interconnection networks I: Sufficient conditions

- MathematicsNetworks
- 2012

This article proves general results regarding the matching preclusion number and the conditional matchingPreclusion number as well as the classification of their respective optimal sets for bipartite graphs.

### Edge proximity conditions for extendability in cubic bipartite graphs

- Mathematics, Computer ScienceJ. Graph Theory
- 2007

We show that a set M of m edges in a cyclically (3m − 2)‐edge‐connected cubic bipartite graph is contained in a 1‐factor whenever the edges in M are pairwise distance at least f(m) apart, where…