185 research outputs found
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
Box representations of embedded graphs
A -box is the cartesian product of intervals of and a
-box representation of a graph is a representation of as the
intersection graph of a set of -boxes in . It was proved by
Thomassen in 1986 that every planar graph has a 3-box representation. In this
paper we prove that every graph embedded in a fixed orientable surface, without
short non-contractible cycles, has a 5-box representation. This directly
implies that there is a function , such that in every graph of genus , a
set of at most vertices can be removed so that the resulting graph has a
5-box representation. We show that such a function can be made linear in
. Finally, we prove that for any proper minor-closed class ,
there is a constant such that every graph of
without cycles of length less than has a 3-box representation,
which is best possible.Comment: 16 pages, 6 figures - revised versio
Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk
Let G be a plane graph of girth at least five. We show that if there exists a
3-coloring phi of a cycle C of G that does not extend to a 3-coloring of G,
then G has a subgraph H on O(|C|) vertices that also has no 3-coloring
extending phi. This is asymptotically best possible and improves a previous
bound of Thomassen. In the next paper of the series we will use this result and
the attendant theory to prove a generalization to graphs on surfaces with
several precolored cycles.Comment: 48 pages, 4 figures This version: Revised according to reviewer
comment
Clique complexes and graph powers
We study the behaviour of clique complexes of graphs under the operation of
taking graph powers. As an example we compute the clique complexes of powers of
cycles, or, in other words, the independence complexes of circular complete
graphs.Comment: V3: final versio
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