Abstract
Let G = ( V(G), E(G )) be a connected graph. Denote by f min( G ) the minimum number of faces taken over all 2-cell embeddings of G into orientable surfaces. For every subset A of E(G ), let p(A ) (resp. i(A )) be the number of components of G/A = (V(G),E(G) – A ) with even (resp. odd) Betti number. Set y A ( G ) = p(A ) + 2 i(A )-| A ∩ E(G ). Using known results on the maximum genus and some matroid theoretical results, L. Nebesky showed that max A y A ( G ) = f min( G ). In this paper, we give another direct proof of this result where only graph-theoretical concepts are used.
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