Abstract
The graphG is called a porcupine, ifG∣A is a complete graph for some setA, every other vertex has degree one, and its only edge is joined toA. In this paper a conjecture of Bollobás is settled almost completely. Namely, it is proved that ifG is a graph onn vertices of diameter 3 with maximum degreeD, D > 2.31\(\sqrt n \),D ≠ (n − 1)/2 and it has the mimimum number of edges, then it is a porcupine.
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Bollobás, B.: Graphs with a given diameter and maximal valency and with a minimal number of edges, in “Comb. Math. and its Appl.” (Welsh, D.J.A., ed.), pp. 25–37, London and New York: Academic Press 1971
Bollobás, B.: Extremal Graph Theory, London and New York: Academic Press 1978
Erdös, P., Rényi, A. and Sós, V.T.: On a problem of graph theory, Stud. Sci. Math. Hung.1, 215–235 (1966)
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This paper was written while the author visited the Departments of Mathematics, Tel-Aviv University, whose hospitality is gratefully acknowledged.
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Füredi, Z. Graphs of diameter 3 with the minimum number of edges. Graphs and Combinatorics 6, 333–337 (1990). https://doi.org/10.1007/BF01787701
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DOI: https://doi.org/10.1007/BF01787701