950 research outputs found
Injective colorings of graphs with low average degree
Let \mad(G) denote the maximum average degree (over all subgraphs) of
and let denote the injective chromatic number of . We prove that
if and \mad(G)<\frac{14}5, then . When
, we show that \mad(G)<\frac{36}{13} implies . In
contrast, we give a graph with , \mad(G)=\frac{36}{13}, and
.Comment: 15 pages, 3 figure
Planar graphs are 9/2-colorable
We show that every planar graph has a 2-fold 9-coloring. In particular,
this implies that has fractional chromatic number at most . This
is the first proof (independent of the 4 Color Theorem) that there exists a
constant such that every planar has fractional chromatic number at
most .Comment: 12 pages, 6 figures; following the suggestion of an editor, we split
the original version of this paper into two papers: one is the current
version of this paper, and the other is "Planar graphs have Independence
Ratio at least 3/13" (also available on the arXiv
Linear Choosability of Sparse Graphs
We study the linear list chromatic number, denoted \lcl(G), of sparse
graphs. The maximum average degree of a graph , denoted \mad(G), is the
maximum of the average degrees of all subgraphs of . It is clear that any
graph with maximum degree satisfies \lcl(G)\ge
\ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if
\mad(G)<12/5 and , then \lcl(G)=\ceil{\Delta(G)/2}+1, and
we give an infinite family of examples to show that this result is best
possible; (2) if \mad(G)<3 and , then
\lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to
show that the bound on \mad(G) cannot be increased in general; (3) if is
planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure
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
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