# An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement

## Abstract

We say that a graph is k-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance at least k in these subsets (i.e., the complement of the kth power of this graph is connected). We say that a graph is k-mono-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance exactly k in these subsets.We prove that the complement of a 3-wide graph on n vertices has at least 3n − 7 edges, and the complement of a 3-mono-wide graph on n vertices has at least 3n − 8 edges. We construct infinite series of graphs for which these bounds are attained.We also prove an asymptotically tight bound for the case k ≥ 4: the complement of a k-wide graph contains at least (n − 2k)(2k − 4[log{sub 2}k] − 1) edges.

- Authors:

- St.Petersburg State University (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22774014

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 1; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; DISTANCE; GRAPH THEORY; PARTITION; VORTICES

### Citation Formats

```
Samoilov, V. S., E-mail: sammarize@gmail.com.
```*An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3860-7.

```
Samoilov, V. S., E-mail: sammarize@gmail.com.
```*An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement*. United States. doi:10.1007/S10958-018-3860-7.

```
Samoilov, V. S., E-mail: sammarize@gmail.com. Sun .
"An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement". United States. doi:10.1007/S10958-018-3860-7.
```

```
@article{osti_22774014,
```

title = {An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement},

author = {Samoilov, V. S., E-mail: sammarize@gmail.com},

abstractNote = {We say that a graph is k-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance at least k in these subsets (i.e., the complement of the kth power of this graph is connected). We say that a graph is k-mono-wide if for any partition of its vertex set into two subsets, one can choose vertices at distance exactly k in these subsets.We prove that the complement of a 3-wide graph on n vertices has at least 3n − 7 edges, and the complement of a 3-mono-wide graph on n vertices has at least 3n − 8 edges. We construct infinite series of graphs for which these bounds are attained.We also prove an asymptotically tight bound for the case k ≥ 4: the complement of a k-wide graph contains at least (n − 2k)(2k − 4[log{sub 2}k] − 1) edges.},

doi = {10.1007/S10958-018-3860-7},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 1,

volume = 232,

place = {United States},

year = {2018},

month = {7}

}