Bled'11 - 7th Slovenian International Conference on Graph Theory

19-25 June 2011
Bled, Slovenia
Europe/Ljubljana timezone
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The chromatic numbers of distance graphs and applications to combinatorial problems.

Presented by Mr. Andrey KUPAVSKII
Type: Oral presentation
Track: Coloring, independence and forbidden subgraphs

Content

We consider \textit{distance graphs in $\mathbb{R}^n$}. The sets of vertices of these graphs are subsets of $\mathbb{R}^n.$ Two vertices are connected by an edge if the distance between them is exactly $d$. This work deals with distance graphs $G=G(n,m,\{a_0, a_1, \ldots, a_m\},x) = (V,E)$ of certain type. Namely, such graphs have the sets of vertices $$V =\{y=(y_1,\ldots,y_n), y_i\in \{0,1,\ldots,m\},$$ $$\ \ \ \ \ \ |\{i:y_i=j\}|=a_jn\ \ \forall j=0,\ldots,m, \ a_i\in(0,1), a_0+a_1+\ldots +a_m=1\}$$ and the sets of edges $$E=\{\{y_1,y_2\}| y_1,y_2\in V, (y_1,y_2)=xn\}.$$ We managed to prove the following theorem. \begin{thm} Let $m=1, a_1=a, x>0, a,x\in \mayhbb{Q},$ and consider the sequence of dimensions $n_1, n_2,\ldots,$ which satisfies the condition $an_i, xn_i\in \mathbb{N}.$ Consider distance graphs $G_i=G(n_i,1,\{1-a,a\},x)$. Let $k$ be a natural number, $k\ge 3$. If $$x<\frac{(ka)^2-\{ka\}^2-[ka]}{k(k-1)}=f_1,$$ then $G_i$ do not contain complete subgraphs (cliques) on $k$ vertices. Moreover, this bound is in some sense sharp. Namely, there exists a constant $c=c(k,a),$ such that in the sequence of dimensions $cn_1, cn_2,\ldots$ graphs $\tilde{G}_i=G(cn_i,1,\{1-a,a\},f_1)$ contain complete subgraphs on $k$ vertices. \end{thm} We proved similar theorems for bigger values of $m$. We applied these results to the following problem. Let $G$ be a distance graph in $\mathbb{R}^n$, that does not contain cliques of size $k$. How large can the chromatic number of such graph be? This problem is related to the classical problem of finding the value $\chi(\mathbb {R}^n)$ which is equal to the minimum number of colors needed to paint all the points in $\mathbb {R}^n$ so that any two points at distance 1 apart receive different colors. It is known, that $$(1.239+o(1))^n\le\chi(\mathbb{R}^n)\le (3+o(1))^n.$$ We obtained new exponential lower bounds for the chromatic number of $k$-clique-free distance graphs for small $k\ge 4$. Another application is related to well-known Borsuk's problem -- the problem of finding the minimum number $f(n)$ of parts of smaller diameter, into which an arbitrary set of diameter 1 in $\mathbb {R}^n$ can be divided. Famous Borsuk's conjecture states that $f(n) = n+1$? We study some modification of Borsuk's problem. Namely, we consider sets of diameter 1, lying on the spheres $S_r^{n-1}$ of radius $r$. Using graphs of the above described type, we managed to prove the following theorem: \noindent {\bf Theorem 1.} {\it Let $S_r^{n-1} \subset {\mathbb R}^n$ be the sphere of radius $r$ with center at the origin. For any $r > \frac{1}{2}$, there exists a $n_0 = n_0(r)$ such that for every $n \ge n_0$, one can find a set $\Omega \subset S_r^{n-1}$ which has diameter 1 and does not admit a partition into $n+1$ parts of smaller diameter.} The work is done under the financial support of the grant 09-01-00294 of the Russian Foundation for Basic Research and the grant MD-8390.2010.1 of the Russian President.

Place

Location: Bled, Slovenia
Address: Best Western Hotel Kompas Bled

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