4,785 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
A combinatorial approach to knot recognition
This is a report on our ongoing research on a combinatorial approach to knot
recognition, using coloring of knots by certain algebraic objects called
quandles. The aim of the paper is to summarize the mathematical theory of knot
coloring in a compact, accessible manner, and to show how to use it for
computational purposes. In particular, we address how to determine colorability
of a knot, and propose to use SAT solving to search for colorings. The
computational complexity of the problem, both in theory and in our
implementation, is discussed. In the last part, we explain how coloring can be
utilized in knot recognition
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
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