Duke's theorem does not extend to signed graph embeddings (Q1182972)

For a connected graph \(G\), \(M(G)\) is the set of all integers \(g\) such that \(G\) has a 3-cell imbedding on the closed orientable 2-manifold \(S_g\) of genus \(g\). A theorem of \textit{R. A. Duke} [Can. J. Math. 18, 817--822 (1966; Zbl 0141.21302)] establishes that \(M(G)\) is always an interval of nonnegative integers. \textit{S. Stahl} [J. Graph Theory 2, 41--52 (1978; Zbl 0396.05013)] established the nonorientable analog. A function \(f\colon E(G)\to\{-1,+1\}\) converts \(G\) into a signed graph. A cycle \((e_1,e_2,\ldots,e_k)\) is balanced if \(\prod^k_{i=1} f(e_i)=+1\). An orientation imbedding of \(G\) is a 2-cell imbedding \(i\colon G\to S\) of a connected signed graph into a closed 2-manifold \(S\) such that a cycle \(C\) of \(G\) is balanced if and only if \(i(C)\) is orientation-preserving on \(S\). For a connected unbalanced signed graph \(G\), \(M_{\text{or}}(G)\) is the set of all integers \(h\) for which there is a 2-cell orientation imbedding of \(G\) on the closed nonorientable 2-manifold \(N_h\) of genus \(h\). The purpose of this note is to disprove a conjecture of Zaslavsky (in analogy with the Duke and Stahl results) by constructing connected unbalanced signed graphs \(G\) whose sets \(M_{\text{or}}(G)\) are arbitrarily large, but contain no pair of consecutive integers. The construction uses homology-type arguments and surgery.