Abstract: In [B], a $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-8$ was constructed. In this article we prove that $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-8$ is unique. As a consequence, we get that the minimum number of edges in a non-orientable polyhedral map of Euler characteristic $-8$ is $> 40$. We have also constructed $\{5, 5\}$-equivelar polyhedral map of Euler characteristic $-2m$ for each $m\geq 4$.
[B] Brehm, U.: Polyhedral maps with few edges. Topics in Comb. and Graph Theory (Ringel-Festschrift) (eds. Bodendiek, R. and Henn, R.), Physica-Verlag, Heidelberg 1990, 153-162.
Keywords: polyhedral maps, polyhedral 2-manifold
Classification (MSC2000): 52B70, 51M20, 57M20
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