Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 583-596 (2002)

The Edge-minimal Polyhedral Maps of Euler Characteristic $-8$

Ulrich Brehm, Basudeb Datta, Nandini Nilakantan

Institut f\"{u}r Geometrie, Technische Universit\"{a}t Dresden,D-01062 Dresden, Germany, e-mail: brehm@math.tu-dresden.de; Department of Mathematics, Indian Institute of Science Bangalore 560 012, India, e-mail: dattab@math.iisc.ernet.in; e-mail: nandini@math.iisc.ernet.in

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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