3 research outputs found
Face distributions of embeddings of complete graphs
A longstanding open question of Archdeacon and Craft asks whether every
complete graph has a minimum genus embedding with at most one nontriangular
face. We exhibit such an embedding for each complete graph except , the
complete graph on 8 vertices, and we go on to prove that no such embedding can
exist for this graph. Our approach also solves a more general problem, giving a
complete characterization of the possible face distributions (i.e. the numbers
of faces of each length) realizable by minimum genus embeddings of each
complete graph. We also tackle analogous questions for nonorientable and
maximum genus embeddings.Comment: 37 pages, 32 figure
Quadrangular embeddings of complete graphs and the Even Map Color Theorem
Hartsfield and Ringel constructed orientable quadrangular embeddings of the
complete graph for , and nonorientable ones for and . These provide minimal quadrangulations of their
underlying surfaces. We extend these results to determine, for every complete
graph , , the minimum genus, both orientable and nonorientable,
for the surface in which has an embedding with all faces of degree at
least , and also for the surface in which has an embedding with all
faces of even degree. These last embeddings provide sharpness examples for a
result of Hutchinson bounding the chromatic number of graphs embedded with all
faces of even degree, completing the proof of the Even Map Color Theorem. We
also show that if a connected simple graph has a perfect matching and a
cycle then the lexicographic product has orientable and nonorientable
quadrangular embeddings; this provides new examples of minimal
quadrangulations.Comment: Version 2 is a greatly expanded version of the paper with four new
authors (Lawrencenko, Chen, Hartsfield, Yang). It includes results on
quadrangular or nearly quadrangular embeddings for all complete graphs. It
also includes an application to a coloring result. This version contains some
details omitted from the version that will be submitted for publication. 26
pages; 6 figure
Minimal quadrangulations of surfaces
A quadrangular embedding of a graph in a surface , also known as a
quadrangulation of , is a cellular embedding in which every face is
bounded by a -cycle. A quadrangulation of is minimal if there is no
quadrangular embedding of a (simple) graph of smaller order in . In
this paper we determine , the order of a minimal quadrangulation of
a surface , for all surfaces, both orientable and nonorientable.
Letting denote the sphere and the Klein bottle, we prove that
, and for all other surfaces , where
is the Euler characteristic. Our proofs use a `diagonal
technique', introduced by Hartsfield in 1994. We explain the general features
of this method.Comment: 25 pages, 20 figure