An Upper Bound on the Number of Edges of a Graph Whose kth Power Has a Connected Complement
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 k th power of this graph is connected). We say that a graph is k -mono-wide if for any partition of its vertex set into...
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Veröffentlicht in: | Journal of mathematical sciences (New York, N.Y.) N.Y.), 2018-07, Vol.232 (1), p.84-97 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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
k
th 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 3
n
− 7 edges, and the complement of a 3-mono-wide graph on
n
vertices has at least 3
n
− 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
− 2
k
)(2
k
− 4[log
2
k
] − 1) edges. |
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ISSN: | 1072-3374 1573-8795 |
DOI: | 10.1007/s10958-018-3860-7 |