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On graphs with equal edge connectivity and minimum degree

✍ Scribed by Béla Bollobás

Publisher
Elsevier Science
Year
1979
Tongue
English
Weight
255 KB
Volume
28
Category
Article
ISSN
0012-365X

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✦ Synopsis

If a grrrph G hao edge connectivity A then the vertex fiat ha a partition V(a) = U U W ash that 61 esntainti exactly A edgea from U to W, Wen~se if Qo ia a maximal graph of order n and edge connectivity A than C$, is sbtctined from the dkjsint union of two complete oubgragh8, B,[U] and &T,[ Wg, by adding A edgea from Ur to W. (Here and in the sequel we use the notation of [I],) we may aaume without IOBB of generality that 1 Ul = k 1 WI = m, The rret U eantaina a vertex that i$ adjacent to at ma@t LA/k] vartic in W; wo S(B,,) Since $(a) B A, this imgliea that one of the fallowing four C~IXS holds:

(1) k = 1, (ii) k = A and every vertex in U 1~ adjacent to exactly one vertex in W, (iii) k = A =t= 1 and B(C3,) = A, (iv) k a A + ii afid 43(<Ta) = k -f 3 Ai h the! Atist three QE~MBI A(@,) y g(g,,) an# &~ce

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