Abstract
Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus γ M ( G) equals ⌊ β( G⧸2⌋, where β( G)=| E( G)|− | V( G)| + 1 is the Betti number of G. For graphs with loops we show that ⌈ β( G)⧸2⌉ − 2⩽ γ M ( G)⩽⌊ β( G)⧸2⌋ if G is vertex 2-connected, and compute the exact value of γ M ( G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [10] 4-connected graphs are upper embeddable.
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