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[log2k] − 1) edges.
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References
F. Harary, Graph Theory, Addison-Wesley (1969).
R. Diestel, Graph Theory, Springer-Verlag, New York (1997).
J. A. Bondy and U. S. R. Murty, Graph Theory With Applications, North-Holland, New York (1997).
H. Fleischner, “The square of every two-connected graph is Hamiltonian,” J. Combin. Theory Ser. B, 16, 29–34 (1974).
A. M. Hobbs, “Some Hamiltonian results in powers of graphs,” J. Res. Natl. Bur. Stand. Sect. B, 77, 1–10 (1973).
G. Chartrand and S. F. Kapoor, “The cube of every connected graph is 1-Hamiltonian,” J. Res. Natl. Bur. Stand. Sect. B, 73, 47–48 (1969).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 151–174.
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Samoilov, V.S. An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement. J Math Sci 232, 84–97 (2018). https://doi.org/10.1007/s10958-018-3860-7
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DOI: https://doi.org/10.1007/s10958-018-3860-7