48 research outputs found
Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces
A method is suggested for construction of quadrangulations of the closed
orientable surface with given genus g and either (1) with given chromatic
number or (2) with given order allowed by the genus g. In particular, N.
Hartsfield and G. Ringel's results [Minimal quadrangulations of orientable
surfaces, J. Combin. Theory, Series B 46 (1989) 84-95] are generalized by way
of generating new minimal quadrangulations of infinitely many other genera.Comment: 6 pages. This version is only slightly different from the original
version submitted on 8 Jul 2012: the author's affiliation has been changed
and the presentation has been slightly improve
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate