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Large planar graphs with given diameter and maximum degree. (English) Zbl 0832.05060

Let \(G\) be a planar graph on \(n\) vertices with maximum degree \(\Delta\) and diameter \(k\). The problem of determining the maximum number of vertices that \(G\) can have is studied in this paper. When \(k= 2\), it has been shown that \(n\leq \lfloor{3\over 2}\Delta\rfloor+ 1\) (for \(\Delta\geq 8\)). In this paper, the authors show that when \(k= 3\), we have \(\lfloor {9\over 2} \Delta\rfloor- 3\leq n\leq 8\Delta+ 12\). They also show that in general \(n\) is \(\Theta(\Delta^{\lfloor k/2\rfloor})\) for any fixed value of \(k\).

MSC:

05C35 Extremal problems in graph theory
05C10 Planar graphs; geometric and topological aspects of graph theory
05C12 Distance in graphs
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