Extension des scalaires par le morphisme de Frobenius,pour les groupes réductifs

Let G be a reductive group over a field k of characteristic $p>0$. For $n?0$ and $q:=p^n$, let $G^{\{n\}}$ be deduced from G by the extension of scalars $x?x^q:k?k$. If k is perfect, this keeps making sense for $n?\mathbb{Z}$. We show that, if k is perfect, there exists $m>0$ such that the algebraic groups G and $G^{\{m\}}$ over k are isomorphic. The isomorphism class of $G^{\{n\}}$, as a reductive group over k, then depends only on n modulo m. For k not necessarily perfect, we show that such a periodicity remains true for n large enough.     

Degree Sequence of Tight Distance Graphs

A graph G on n vertices is a tight distance graph if there exists a set $D\subseteq \{1,2,\dots ,n-1\}$ such that $V(G)=\{0,1,\dots ,n-1\}$ and $ij\in E(G)$ if and only if $|i-j|\in D$. A characterization of the degree sequences of tight distance graphs is given. This characterization yields a fast method for recognizing and realizing degree sequences of tight distance graphs.     

Solvable PST-groups,strong Sylow bases and mutually permutable products

A subgroup H of a group G is said to permute with the subgroup K of G if $HK=KH$. Subgroups H and K are mutually permutable (totally permutable) in G if every subgroup of H permutes with K and every subgroup of K permutes with H (if every subgroup of H permutes with every subgroup of K). If H and K are mutually permutable and $H\cap K=1$, then H and K are totally permutable. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a PST-group if S-permutability is a transitive relation in G. Let $\{p_1,\dots ,p_n,p_{n+1},\dots ,p_k\}$ be the set of prime divisors of the order of a finite group G with $\{p_1,\dots ,p_n\}$ the set of prime divisors of the order of the normal subgroup N of G. A set of Sylow subgroups $\{P_1,\dots ,P_n,P_{n+1},\dots ,P_k\}$, $P_i\in {\mathrm{Syl}}_{p_i}(G)$, form a strong Sylow system with respect to N if $P_i{P}_j$ is a mutually permutable product for all $i\in \{1,2,\dots ,n\}$ and $j\in \{1,2,\dots ,k\}$. We show that a finite group G is a solvable PST-group if and only if it has a normal subgroup N such that $G/N$ is nilpotent and G has a strong Sylow system with respect to N. It is also shown that G is a solvable PST-group if and only if G has a normal solvable PST-subgroup N and $G/N^{″}$ is a solvable PST-group.     

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number $\mathrm{adrn}(G)$ is the least k such that every set of k dacards determines G. We determine $\mathrm{adrn}(D_{m,n,p})$, where the double-broom $D_{m,n,p}$ with $p\ge 2$ is the tree with $m+n+p$ vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine $\mathrm{adrn}(D_{m,n,p})$ for all $m,n,p$. For $2\le m\le n$, usually $\mathrm{adrn}(D_{m,n,p})=m+2$, except $\mathrm{adrn}(D_{m,m+1,p})=m+1$ and $\mathrm{adrn}(D_{m,m+2,p})=m+3$. There are exceptions when $(m,n)=(2,3)$ or $p=4$. For $m=1$ the usual value is 4, with exceptions when $p\in \{2,3\}$ or $n=2$.     

Fractional parts of powers and Sturmian words

Let $b?2$ be an integer. In terms of combinatorics on words we describe all irrational numbers $\xi >0$ with the property that the fractional parts $\{\xi b^n\}$, $n?0$, all belong to a semi-open or an open interval of length $1/b$. The length of such an interval cannot be smaller, that is, for irrational ξ, the fractional parts $\{\xi b^n\}$, $n?0$, cannot all belong to an interval of length smaller than $1/b$. To cite this article: Y. Bugeaud, A. Dubickas, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Coloring circle graphs

Circle graph is an intersection graph of chords of a circle. We denote the class of circle graphs by cir. In this paper we investigate the chromatic number of the circle graph as a function of the size of maximum clique $\omega =\omega (G)$. More precisely we investigate $f(k)=\max \{\chi (G)|G\in \text{CIR }\mathrm{\&}\omega (G)⩽k\}$. Kratochvíl and Kostochka showed that $f(k)⩽50\cdot 2^k-32k-64$. The best lower bound is by Kostochka and is $f(k)=\mathrm{\Omega }(k\log k)$. We improve the upper bound to $f(k)⩽21\cdot 2^k-24k-24$. We also present the bound $\chi (G)⩽\omega \cdot \log n$ which shows, that the circle graphs with small maximum clique and large chromatic number must have many vertices.