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Three-coloring triangle-free graphs on surfaces III. Graphs of girth five
We show that the size of a 4-critical graph of girth at least five is bounded
by a linear function of its genus. This strengthens the previous bound on the
size of such graphs given by Thomassen. It also serves as the basic case for
the description of the structure of 4-critical triangle-free graphs embedded in
a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark
Spheres are rare
We prove that triangulations of homology spheres in any dimension grow much
slower than general triangulations. Our bound states in particular that the
number of triangulations of homology spheres in 3 dimensions grows at most like
the power 1/3 of the number of general triangulations.Comment: 14 pages, 1 figur
3-coloring triangle-free planar graphs with a precolored 8-cycle
Let G be a planar triangle-free graph and let C be a cycle in G of length at
most 8. We characterize all situations where a 3-coloring of C does not extend
to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure
Asymptotic enumeration and limit laws for graphs of fixed genus
It is shown that the number of labelled graphs with n vertices that can be
embedded in the orientable surface S_g of genus g grows asymptotically like
where , and is the exponential growth rate of planar graphs. This generalizes the
result for the planar case g=0, obtained by Gimenez and Noy.
An analogous result for non-orientable surfaces is obtained. In addition, it
is proved that several parameters of interest behave asymptotically as in the
planar case. It follows, in particular, that a random graph embeddable in S_g
has a unique 2-connected component of linear size with high probability
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