Fellows, M.; Hell, P.; Seyffarth, K. Large planar graphs with given diameter and maximum degree. (English) Zbl 0832.05060 Discrete Appl. Math. 61, No. 2, 133-153 (1995). 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 \(k\). Reviewer: Z.Chen (Indianapolis) Cited in 1 ReviewCited in 15 Documents MSC: 05C35 Extremal problems in graph theory 05C10 Planar graphs; geometric and topological aspects of graph theory 05C12 Distance in graphs Keywords:planar graph; maximum degree; diameter PDFBibTeX XMLCite \textit{M. Fellows} et al., Discrete Appl. Math. 61, No. 2, 133--153 (1995; Zbl 0832.05060) Full Text: DOI References: [1] Bermond, J-C.; Delorme, C.; Quisquater, J.-J., Strategies for interconnection networks: some methods from graph theory, J Parallel Distributed Comput., 3, 433-449 (1986) [2] (Bermond, J-C., DAMIN, Special double volume on interconnection networks. DAMIN, Special double volume on interconnection networks, Discrete Appl. Math., 37 (1992)) [3] Delorme, C., Large bipartite graphs with given degree and diameter, J. Graph Theory, 9, 325-334 (1985) · Zbl 0623.05048 [4] M. Fellows, P. Hell and K. Seyffarth, Constructions of dense planar networks, preprint.; M. Fellows, P. Hell and K. Seyffarth, Constructions of dense planar networks, preprint. · Zbl 1002.90008 [5] Hell, P.; Seyffarth, K., Largest planar graphs of diameter two and fixed maximum degree, Discrete Math., 111, 313-332 (1993) · Zbl 0837.05074 [6] Lipton, R. J.; Tarjan, R. E., A separator theorem for planar graphs, SIAM J. Appl. Math., 36, 177-189 (1979) · Zbl 0432.05022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.