IJPAM: Volume 67, No. 1 (2011)

A NOTE ON UPPER BOUND FOR $d$-COVERED
TRIANGULATION OF CLOSED SURFACES

Ashish K. UpadhyayDepartment of Mathematics, Indian Institute of Technology Patna, Patliputra Colony, Patna, 800013, INDIA
School of Basic and Applied Sciences
Guru Gobind Singh Indraprastha University
Kashmere Gate, Delhi, 110006, INDIA
e-mails: [email protected], [email protected]


Abstract.A triangulation of a closed surface is said to be $d$-covered if at least one vertex in each edge has degree $d$. In [#!NeNa!#] the authors have shown that: (a) if a closed surface of non-positive Euler characteristic $\chi$ admits a $d$-covered triangulation then $d \leq 2\,\lfloor \frac{5+ \sqrt{49 - 24\,\chi}}{2}\rfloor$, and (b) the upper bound for $d$ is attained when: $(i)\,\chi = 0$ and for $(ii)\, d = 2 (n - 1)$, whenever $n = \frac{7 + \sqrt{49 - 24\,\chi}}{2}$ is a positive integer. In this note we show that the upper bound is attained for $\chi \in S:=\{-2, \ldots, -8\}\cup\{-13, -14, -15\}$.

Received: September 14, 2008

AMS Subject Classification: 57Q15, 52B70

Key Words and Phrases: triangulations of surfaces, equivelar and $d$-covered triangulations

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2011
Volume: 67
Issue: 1