16,355,142 research outputs found
Large restricted sumsets in general abelian group
Let A, B and S be three subsets of a finite Abelian group G. The restricted
sumset of A and B with respect to S is defined as A\wedge^{S} B= {a+b: a in A,
b in B and a-b not in S}. Let L_S=max_{z in G}| {(x,y): x,y in G, x+y=z and x-y
in S}|. A simple application of the pigeonhole principle shows that
|A|+|B|>|G|+L_S implies A\wedge^S B=G. We then prove that if |A|+|B|=|G|+L_S
then |A\wedge^S B|>= |G|-2|S|. We also characterize the triples of sets (A,B,S)
such that |A|+|B|=|G|+L_S and |A\wedge^S B|= |G|-2|S|. Moreover, in this case,
we also provide the structure of the set G\setminus (A\wedge^S B).Comment: Paper submitted November 15, 2011. To appear European Journal of
Combinatorics, special issue in memorian Yahya ould Hamidoune (2013
Improved bounds on coloring of graphs
Given a graph with maximum degree , we prove that the
acyclic edge chromatic number of is such that . Moreover we prove that:
if has girth ; a'(G)\le
\lceil5.77 (\Delta-1)\rc if
has girth ; a'(G)\le \lc4.52(\D-1)\rc if ;
a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil.
We further prove that the acyclic (vertex) chromatic number of is
such that
a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the
star-chromatic number of is such that \chi_s(G)\le
\lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic
number \chi^\b(G) of is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\;
k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are
decreasing functions of \b such that k_1(\b)\in[4, 6] and
k_2(\b)\in[2,5].
To obtain these results we use an improved version of the Lov\'asz Local
Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of
Theorem 2 (items c-f) written in more detail
Homogeneous products of conjugacy classes
Let be a finite group and . Let be
the conjugacy class of in . Assume that and are conjugacy
classes of with the property that . Then is a conjugacy class if and only if and is a
normal subgroup of
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