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[3] viXra:2205.0066 [pdf] submitted on 2022-05-12 23:49:13
Authors: Theophilus Agama
Comments: 8 Pages.
Using the method of compression we show that the number of integral points in the annular region induced by two $k$ dimensional spheres of radii $r$ and $R$ with $R>r$ satisfies the lower bound
\begin{align} \mathcal{N}_{R,r,k} \gg (R^{k-1}-r^{k+\delta})\sqrt{k}.\nonumber
\end{align}for some small $\delta>0$ with $k>\frac{\delta(\log r)}{\log R-\log r}$.
Category: Combinatorics and Graph Theory
[2] viXra:2205.0049 [pdf] submitted on 2022-05-09 15:36:12
Authors: Doron Zeilberger
Comments: 1 Page.
Inspired by Mark Levi's wonderful proof of the Cosine addition formula, that showed that it follows from the sad fact that Perpetual Motion is impossible, we recall five other proofs.
Category: Combinatorics and Graph Theory
[1] viXra:2205.0002 [pdf] submitted on 2022-05-01 21:50:47
Authors: Elmar Guseinov
Comments: Pages.
The Erdős-Gyárfás conjecture (EGC) states that every graph with minimum vertex degree of at least 3 contains a cycle whose length is a power of 2. The statement has not been proven even for cubic graphs. Moreover, it has not been proven for cubic Hamiltonian graphs. On the other hand, we can see that every cubic Hamiltonian graph has an even number of vertices 2k and can be obtained by adding k edges to a 2k-cycle. It will be shown that adding this number of edges to a given vertex of the cycle always gives a 4-cycle. This suggests the validity of EGC for cubic Hamiltonian graphs. This theorem is a consequence of a more general result the proof of which is the main content of the paper. Namely, we will determine the maximum number of edges the addition of which to a given vertex of a cycle does not give a cycle of a sufficiently small length.
Category: Combinatorics and Graph Theory
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