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On incompactness for chromatic number of graphs
set theory graphs chromatic number compactness non-reflecting stationary sets
2012/5/9
We deal with incompactness. Assume the existence of non-reflecting stationary set of cofinality kappa . We prove that one can define a graph G whose chromatic number is > kappa, while the chromatic nu...
Unknotting number and number of Reidemeister moves needed for unlinking
link diagram Reidemeister move Hass-Nowik knot diagram invariant
2011/2/22
Using unknotting number, we introduce a link diagram invariant of Hass and Nowik type, which changes at most by 2 under a Reidemeister move. As an application,we show that a certain infinite sequence ...
Nordhaus-Gaddum-type theorem for rainbow connection number of graphs
edge-colored graph rainbow connection number Nordhaus-Gaddumtype
2011/1/21
An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of G, denoted rc(G), is the minimum number of c...
Gaussian Relay Channel Capacity to Within a Fixed Number of Bits
Gaussian Relay Channel Capacity Fixed Number of Bits
2010/11/24
In this paper, we show that the capacity of the three-node Gaussian relay channel can be achieved to within 1 and 2 bit/sec/Hz using compress-and-forward and amplify-and-forward relaying, respectively...
Determination of the number of isomorphism classes of extensions of a $\kp$-adic field
extensions $\kp$-adic field
2010/11/8
We deduce a formula enumerating the isomorphism classes of extensions of a $\kp$-adic field $K$ with given ramification $e$ and inertia $f$. The formula follows from a simple group-theoretic lemma, pl...
Notes on factor-criticality, extendibility and independence number
Notes on factor-criticality independence number
2010/11/19
In this paper, we give a sufficient and necessary condition for a $k$-extendable graph to be $2k$-factor-critical when $k=\nu/4$, and prove some results on independence numbers in $n$-factor-critical ...
The number of Goldbach representations of an integer
Goldbach representations of an integer math
2010/11/19
We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log...
The Fractional Chromatic Number of Triangle-free Graphs with $\Delta\leq 3$
Triangle-free Graphs Combinatorics
2010/11/18
Let $G$ be any triangle-free graph with maximum degree $\Delta\leq 3$. Staton proved that the independence number of $G$ is at least $\frac{5}{14}n$. Heckman and Thomas conjectured that Staton's resul...
We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative qu...
Residue classes containing an unexpected number of primes
Residue classes unexpected number of primes
2010/12/6
We fix a non-zero integer a and consider arithmetic progressions a mod q, with q varying over a given range. We show that for certain specific values of a, the arithmetic progressions a mod q contain,...
A note on knots with H(2)-unknotting number one
H(2)-unknotting number Goeritz matrix Ozsv´ ath-Szab´ o correction term Heegaard Floer homology
2010/12/8
We give an obstruction to unknotting a knot by adding a twisted band,derived from Heegaard Floer homology.