A note on approximating graph genus

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This note, using simple combinatorial analysis, shows two interesting facts on the approximability of graph minimum genus embeddings: 1. (1) for any function ƒ(n) = O(n ε) , 0 ⩽ ε < 1, there is no polynomial time algorithm that embeds a graph G of n vertices into a surface of genus bounded by γ min(G) + ƒ(n) , unless P = NP; and 2. (2) there is a linear time algorithm that embeds a graph G of n vertices into a surface of genus bounded by max{4 γ min ( G), γ min ( G) + 4 n}, where γ min ( G) denotes the minimum genus of the graph G. An approximation algorithm with approximation ratio O(√ n) for bounded degree graph embeddings is also presented.