1,064 research outputs found
Edge-coloring via fixable subgraphs
Many graph coloring proofs proceed by showing that a minimal counterexample
to the theorem being proved cannot contain certain configurations, and then
showing that each graph under consideration contains at least one such
configuration; these configurations are called \emph{reducible} for that
theorem. (A \emph{configuration} is a subgraph , along with specified
degrees in the original graph for each vertex of .)
We give a general framework for showing that configurations are reducible for
edge-coloring. A particular form of reducibility, called \emph{fixability}, can
be considered without reference to a containing graph. This has two key
benefits: (i) we can now formulate necessary conditions for fixability, and
(ii) the problem of fixability is easy for a computer to solve. The necessary
condition of \emph{superabundance} is sufficient for multistars and we
conjecture that it is sufficient for trees as well, which would generalize the
powerful technique of Tashkinov trees.
Via computer, we can generate thousands of reducible configurations, but we
have short proofs for only a small fraction of these. The computer can write
\LaTeX\ code for its proofs, but they are only marginally enlightening and can
run thousands of pages long. We give examples of how to use some of these
reducible configurations to prove conjectures on edge-coloring for small
maximum degree. Our aims in writing this paper are (i) to provide a common
context for a variety of reducible configurations for edge-coloring and (ii) to
spur development of methods for humans to understand what the computer already
knows.Comment: 18 pages, 8 figures; 12-page appendix with 39 figure
5-list-coloring planar graphs with distant precolored vertices
We answer positively the question of Albertson asking whether every planar
graph can be -list-colored even if it contains precolored vertices, as long
as they are sufficiently far apart from each other. In order to prove this
claim, we also give bounds on the sizes of graphs critical with respect to
5-list coloring. In particular, if G is a planar graph, H is a connected
subgraph of G and L is an assignment of lists of colors to the vertices of G
such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G
contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer
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