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Title: 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:
 [1]
  1. 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}
}