2,377 research outputs found
Third case of the Cyclic Coloring Conjecture
The Cyclic Coloring Conjecture asserts that the vertices of every plane graph
with maximum face size D can be colored using at most 3D/2 colors in such a way
that no face is incident with two vertices of the same color. The Cyclic
Coloring Conjecture has been proven only for two values of D: the case D=3 is
equivalent to the Four Color Theorem and the case D=4 is equivalent to
Borodin's Six Color Theorem, which says that every graph that can be drawn in
the plane with each edge crossed by at most one other edge is 6-colorable. We
prove the case D=6 of the conjecture
Cyclic Coloring of Plane Graphs with Maximum Face Size 16 and 17
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected
plane graph with maximum face size D can be colored using at most D+2 colors in
such a way that no face is incident with two vertices of the same color. The
conjecture has been proven for D=3, D=4 and D>=18. We prove the conjecture for
D=16 and D=17
New Lower Bounds for van der Waerden Numbers Using Distributed Computing
This paper provides new lower bounds for van der Waerden numbers. The number
is defined to be the smallest integer for which any -coloring
of the integers admits monochromatic arithmetic progression of
length ; its existence is implied by van der Waerden's Theorem. We exhibit
-colorings of that do not contain monochromatic arithmetic
progressions of length to prove that . These colorings are
constructed using existing techniques. Rabung's method, given a prime and a
primitive root , applies a color given by the discrete logarithm base
mod and concatenates copies. We also used Herwig et al's
Cyclic Zipper Method, which doubles or quadruples the length of a coloring,
with the faster check of Rabung and Lotts. We were able to check larger primes
than previous results, employing around 2 teraflops of computing power for 12
months through distributed computing by over 500 volunteers. This allowed us to
check all primes through 950 million, compared to 10 million by Rabung and
Lotts. Our lower bounds appear to grow roughly exponentially in . Given that
these constructions produce tight lower bounds for known van der Waerden
numbers, this data suggests that exact van der Waerden Numbers grow
exponentially in with ratio asymptotically, which is a new conjecture,
according to Graham.Comment: 8 pages, 1 figure. This version reflects new results and reader
comment
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
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