Abstract
It is known that for each d there exists a graph of diameter two and maximum degree d which has at least ⌈(d/2)⌉ ⌈(d + 2)/2⌉ vertices. In contrast with this, we prove that for every surface S there is a constant ds such that each graph of diameter two and maximum degree d ≥ ds, which is embeddable in S, has at most ⌊(3/2)d⌋ + 1 vertices. Moreover, this upper bound is best possible, and we show that extremal graphs can be found among surface triangulations. © 1997 John Wiley & Sons, Inc.
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