Surface embedding of non-bipartite $k$-extendable graphs

We find the minimum number $k=μ'(Σ)$ for any surface $Σ$, such that every $Σ$-embeddable non-bipartite graph is not $k$-extendable. In particular, we construct the so-called bow-tie graphs $C_6\bowtie P_n$, and show that they are $3$-extendable. This confirms the existence of an infinite number of $3$-extendable non-bipartite graphs which can be embedded in the Klein bottle.

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