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Extremal graphs of diameter two and given maximum degree, embeddable in a fixed surface
Author(s) -
Knor Martin,
Širáň Jozef
Publication year - 1997
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/(sici)1097-0118(199701)24:1<1::aid-jgt1>3.0.co;2-v
Subject(s) - mathematics , combinatorics , degree (music) , graph , upper and lower bounds , surface (topology) , discrete mathematics , geometry , mathematical analysis , physics , acoustics
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 d s such that each graph of diameter two and maximum degree d ≥ d s , 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.