OSCAR ORDAZ, MAR´IA TERESA VARELA and FELICIA VILLARROEL
Let G be a finite Abelian group of order n. A k-sequence in G is said to be barycentric if it contains an element which is the “average” of its terms. Thek- barycentric Olson constantBO(k, G) is introduced as the minimal positive integer tsuch that anyt-set inGcontains ak-barycentric set. Conditions for the existence of BO(k, G) are established and some values or bounds are given. Moreover, new values for thek-barycentric Davenport constantBD(k, G) and barycentric Ramsey numbers for starsBR(K1,k, G) are given. These constants, defined in some of our previous works, use sequences instead of sets.
AMS 2000 Subject Classification: 05C55, 05C65, 05D10.
Key words: k-barycentric sequence,k-barycentric, k-barycentric Olson constant, barycentric Ramsey number, zero-sum.
1. INTRODUCTION
LetGbe an Abelian group of ordern. Weighted sequences, i.e., sequences constituted by terms of the formwiai, whereaiare elements ofGand the coef- ficients or weights are positive integers, appear initially in the Caro conjecture [2] formulated in 1996.
Conjecture 1.1. Let w1, w2, . . . , wk be positive integers such that w1+ w2+· · ·+wk= 0 (mod n). Leta1, a2, . . . , an+k−1inGnot necessarily distinct.
Then there exist k distinct indices i1, . . . , ik such that w1ai1 +w2ai2 +· · ·+ wkaik = 0.
Hamidoune [11] gave a solution to Caro conjecture under the additional assumption (wi, n) = 1∀i. Recently, Grynkiewicz [10] gave a complete solution to this conjecture in
Theorem 1.2 ([10]). Let m, n and k≥2 be positive integers. If f is a sequence of n+k−1 elements from a nontrivial Abelian group G of order n and exponent m, and if W ={wi}ki=1 is a sequence of integers whose sum is zero modulo m, then there exists a rearranged subsequence {bi}ki=1 of f such that
k
P
i=1
wibi= 0.Furthermore, if f has a k-set partition A=A1, . . . , Ak such
MATH. REPORTS11(61),1 (2009), 33–45
that|wiAi|=|Ai|for all i, then there exists a nontrivial subgroup H of Gand a k-set partition A1 =A11, . . . , A1k of f with H ⊆
k
P
i=1
wiA1i and |wiA1i|=|A1i| for all i.
A weighted k-sequence, k ≥ 2, with terms wiai, wi = 1, 1 ≤ i ≤ k, excepting some j for which wj = 1−k, ai ∈ G and zero-sum, is called a k-barycentric sequence. That is to say, a k-sequenceai, 1≤i≤k,withk≥2, where there exists an aj such that a1+a2+· · ·+aj+· · ·+ak=kaj.
The study of barycentric sequences started in [6] and [7]. In [6] the Davenport barycentric constantBD(G) and in [7] the Davenportk-barycentric constant BD(k, G) are introduced. They are defined as the smallest positive integert such that every sequence of lengthtcontains a barycentric sequence or a k-barycentric sequence, respectively.
LetH= (V(H), E(H)) be a graph withe(H) edges. In [7] the barycen- tric Ramsey number BR(H, G) is defined as the minimum positive integer t such that any coloring c : E(Kt) → G of the edges of Kt by elements of G yields a copy of H, sayH0, with an edgee0 such that
(1) X
e∈E(H0)
c(e) =kc(e0)
In this caseHis called abarycentricgraph. This constant is a generalization of the Ramsey zero-sum numberR(H, G) defined whene(H) = 0 (mod n), as the minimal positive integerssuch that any coloringc:E(Ks)→Gof the edges of Ksby elements ofGcontains a copy ofH, sayH, with P
e∈E(H)
c(e) = 0, where 0 is the zero element ofG. The necessity of the conditione(H) = 0 (mod n) for the existence ofR(H, G) is clear. It comes from the monochromatic coloration of the edges of H.
The main goals of this paper are:
– To define of the k-barycentric Olson constant BO(k, G) and study its existence. The Abelian groups that will be studied areG=Znfor some prime or composite n, in particular for 3≤n≤12 and 3≤k≤n. For those k and n for which BO(k, Zn) exists, this value or bounds of it are given.
– To establish new values forBD(k, G) from BO(k, G).
– To give new values for BR(K1,k, G) from BD(k, G).
Besides this introduction and the conclusion, the paper contains three main sections. Section 2 presents the tools that are used in Sections 3 and 4.
Section 3 is devoted to a study of the k-barycentric Olson constantBO(k, G) for some k and G. In Section 4, new values of BD(k, G) using the values of
BO(k, G) are established. Moreover, new values ofBR(K1,k, G) fromBD(k, G) are given.
2. TOOLS
The following definition is introduced in [6] and [7] and constitutes a natural extension of zero-sum sequences.
Definition 2.1. LetA be a finite set with|A| ≥2 andGa finite Abelian group. A sequencef :A→Gis said to bebarycentricif there existsa∈Asuch that P
x∈A
f(x) =|A|f(a). The element f(a) is called barycenter. The sequence is said to be k-barycentric when|A|=k. Moreover, whenf is injective we use barycentric set instead of barycentric sequence.
Hamidoune [11] gave the following condition.
Hamidoune-condition. Let G be an Abelian group of order n ≥ 2 and f :A→Ga sequence with|A| ≥n+k−1. Then there exists ak-barycentric subsequence of f. Moreover, in the case where k ≥ |G|, the condition |A| ≥ k+D(G)−1, where D(G) is the Davenport constant, is sufficient for the existence of a k-barycentric subsequence of f.
Definition 2.2. Let G be an Abelian group of order n ≥ 2. The k- barycentric Davenport constant BD(k, G) is the minimal positive integer t such that every t-sequence inG contains ak-barycentric subsequence.
By Hamidoune-condition, we haveBD(k, G)≤n+k−1.
In what follows, we present some results about orbits. More information is available in [14].
The set Gn ={fa,b :Zn → Zn, fa,b(x) =ax+b, a, b ∈ Zn,(a, n) = 1}
is a group of order nφ(n) where φ(n) = |{0 < q < n : (q, n) = 1}| is the Euler phi-function. Let Xnk = {{x1, x2, . . . , xk} : xi ∈ Zn}. An action of Gn on Xnk is defined as fa,b({x1, . . . , xk}) ={fa,b(x1), . . . , fa,b(xk)}.It is easy to see that θ({0}) is the only orbit of Xn1 and that the orbits of Xn2 are of the form θ({0, z}).
The Bezout theorem [5] and the Chinese remainder theorem allow to formulate
Lemma 2.3. θ({0, z}) =θ({0, t}), with t= (z, n). Moreover, each orbit in Xn2 contains one and only one{0, t}witht|nandt < n. There are as many orbits in Xn2 as there are divisors of n in {1,2, . . . , n−1}.
Remark 2.4. The orbits of Xnk can be obtained considering for each orbit θ({x1, . . . , xk−1}) of Xnk−1 the sets {x1, . . . , xk−1, xk} with xk ∈ Zn\ {x1, . . . , xk−1}. The action ofGn on these sets defines its orbits.
We have the easy lemmas below.
Lemma 2.5. If {x1, . . . , xk} is a k-barycentric set, then all the elements ofθ({x1, . . . , xk})arek-barycentric. Such an orbit is called barycentric. More- over, if {x1, . . . , xk}contains at-barycentric set, then every element of θ({x1, . . . , xk}) also contains at-barycentric set.
Lemma2.6. The number of orbits ofXnk is equal to the number of orbits of Xnn−k.
Proof. If x, y are in the same orbit of Xnk, then Zn\x and Zn\y also are in the same orbit of Xnn−k. Consequently, the bijection between the sets Xnk and Xnn−k originates a bijection between the orbits ofXnk andXnn−k.
We also need the Dias da Silva-Hamidoune theorem below.
Theorem 2.7 ([8]). Let H be a subset of Zp. Let d be an integer such that 2 ≤ d ≤ |H|. Set ∧dH =
n P
x∈S
x : S ⊂ H,|S| = d o
. Then | ∧dH| ≥ min{p, d(|H| −d) + 1}.
We shall also use
Lemma 2.8 (Hamidoune [12]). Let A be a subset of Zn such that |A| ≥
n+3
2 . Then ∧2A=Zn.
In what follows we present some results about decomposing a complete graph into edge-disjoint subgraphs.
Theorem2.9 (Harary [15]). Let Kn be a complete graph withnvertices.
Then Kn, with n odd, is the edge-disjoint union of n−12 Hamiltonian cycles while Kn, with n even, is the edge-disjoint union of n−22 Hamiltonian cycles and one perfect matching. Hence Kn can be decomposed into n−1 perfect matchings.
Corollary 2.10. Let Kn be a complete graph with n vertices, with n odd. Then Kn can be decomposed into two complete graphs Kn+1
2 sharing a vertex and a bipartite complete graph Kn−1
2 ,n−12 .
Corollary 2.11. Let Kn be a complete graph with n vertices, with n even. Then Kn can be decomposed into two vertex-disjoint complete graphs Kn
2 and the remaining Kn
2,n2 into one perfect matching and one (n2 −1)- regular graph.
Moreover, the following result is used to give the value ofBR(K1,m, Zn) with m= 0 (mod n).
Theorem 2.12 (Bialostocki [1], Caro [4]). Let K1,m be the stars on m edges with m= 0 (modn).Then
BR(K1,m, Zn) =R(K1,m, Zn) =
m+n−1 ifm=n= 0 (mod 2) m+n otherwise.
From now on,p will be a prime number.
3. k-BARYCENTRIC OLSON CONSTANT
Let G be an Abelian group of order n ≥ 3. We establish the existence of BO(k, G) any derive some values or bounds. The existence ofBO(k, Zp) is established in Corollary 3.4 and Remark 3.2.
Definition 3.1. Let G be an Abelian group of order n≥3. The k-bary- centric Olson constant BO(k, G) is the minimal positive integer t such that every t-set inGcontains a k-barycentric set, provided such an integer exists.
Remark 3.2. It is clear that when n is odd, BO(n, Zn) = n while if n is even, BO(n, Zn) does not exist. Moreover, for n odd, the barycenter of every Q ⊆ Zn with |Q| = n−1 is Zn\Q. Therefore, BO(n−1, Zn) does not exist. For n even, BO(n−1, Zn) = n−1. Let S ⊆Zn with |S|=n−1 and {b} = Zn\S.It is easy to see that S is an (n−1)-barycentric set with
n
2 − n−11 b as barycenter.
Theorem 3.3 ([6]). Let s≥ 2, d≥2, p ≥d+ 2 + d−11 . Let A be a set with s+d elements, and f :A → Zp a sequence with |f(A)| ≥ p−1d +d+ 1.
Then f contains an s-barycentric subsequence.
Corollary 3.4. BO(k, Zp)≤p for 3≤k≤p−2 with p≥5.
Theorem 3.5. BO(k, Zn)≤nfor n≥6 and n+12 ≤k≤n−2.
Proof. LetA⊆Znsuch that|A|=k+1. Then|A| ≥ n+32 . By Lemma 2.8 there exists {u, v} ⊆A such thatu+v =P
A
x−ka+a, for somea∈Zn\A.
Then (A\ {u, v})∪ {a} is ak-barycentric set.
Lemma 3.6 ([7]). If a 3-sequence in Zn, n odd, is barycentric, then its elements are equal or pairwise different. Moreover, a3-set inZn is barycentric if and only if its elements are in arithmetic progression.
From this lemma we derive
Remark 3.7. BO(3, Zn)≤n forn≥3.
The following result follows from the Dias da Silva-Hamidoune theorem.
Theorem 3.8. BO(3, Zp)≤ dp3e+ 1for p≥5.
Proof. LetAbe a set inZpwith|A|=dp3e+1. By the theorem of Dias da Silva-Hamidoune we have|∧2A| ≥min{p,2(|A|−2)+ 1}= min{p,2dp3e−1}= 2dp3e −1. Set D={2x :x ∈A}. Then | ∧2A|+|D| ≥ 3dp3e > p. Therefore, there exist 2x∈Dandy+z∈ ∧2Asuch that 2x=y+z, hencex+y+z= 3x, i.e., {x, y, z} is a 3-barycentric set ofA.
In particular, we deduce that –BO(3, Z5) = 3;
– the set {0,4,6} shows that BO(3, Z7) ≥ 4 so, by Theorem 3.8, BO(3, Z7) = 4;
– the set{1,2,4,5}shows that BO(3, Z11)≥5, henceBO(3, Z11) = 5;
– the set{0,1,3,9} shows thatBO(3, Z13)≥5 and, by Theorem 3.8 we have BO(3, Z13)≤6,upper bound improved in the next result.
Lemma 3.9. BO(3, Z13) = 5.
Proof. The action of group G13 on X133 partitions it into three orbits:
θ({0,1,2}) (barycentric) andθ({0,1,3}),θ({0,1,4)}) (non barycentric). Now, we apply Remark 2.4 to the orbitsθ({0,1,3}),θ({0,1,4)}) in order to obtain 3- barycentric-free orbits in X134 . These orbits are: θ({0,1,3,4}), θ({0,1,3,9}), θ({0,1,3,11}), θ({0,1,4,5}), θ({0,1,4,6}), θ({0,1,4,10}). Finally, none of these orbits can be extended by a fifth element without forming a 3-barycentric subset.
Lemma 3.10. BO(3, Zn) = 5 for n= 6,8,9,10.
Proof. n= 6. Under the action of group G6, the set X63 has the follow- ing partition by orbits: θ({0,1,2}), θ({0,2,4}) (barycentric) and θ({0,1,3}) (non barycentric). The only 3-barycentric-free orbit in X64 obtained from θ({0,1,3}), is θ({0,1,3,4}), thus BO(3, Z6) > 4. The only orbit in X65 ob- tained from θ({0,1,3,4}) contains a 3-barycentric set. ThenBO(3, Z6)≤5.
n= 8. Under the action of groupG8, the setX83 has the following parti- tion by orbits: θ({0,1,2}),θ({0,2,4}) (barycentric) andθ({0,1,3}),θ({0,1,4}) (non barycentric). The only 3-barycentric-free orbits in X84 obtained from the non barycentric orbits in X83, are θ({0,1,3,4}) and θ({0,1,4,5}), thus BO(3, Z8) > 4. None of these orbits can be extended by a fifth element in order to form 3-barycentric-free orbits in X85. Then BO(3, Z8)≤5.
n = 9. Under the action of group G9, the set X93 has the following partition by orbits: θ({0,1,2}),θ({0,3,6}) (barycentric) andθ({0,1,6}) (non barycentric). The only 3-barycentric-free orbits inX94obtained fromθ({0,1,6}), areθ({0,1,3,4}) andθ({0,1,3,7}). ThusBO(3, Z9)>4. None of these orbits can be extended by a fifth element in order to form 3-barycentric-free orbits in X95. Then BO(3, Z9)≤5.
n = 10. Under the action of group G10, the set X103 has the follow- ing partition by orbits: θ({0,1,2}), θ({0,2,4}) (barycentric) andθ({0,1,3}), θ({0,1,5}) (non barycentric). The only 3-barycentric-free orbits in X104 ob- tained from the non barycentric orbits in X103 areθ({0,1,3,4}),θ({0,1,3,8}), θ({0,1,4,5}) and θ({0,1,5,6}). Thus BO(3, Z10) > 4. None of these orbits can be extended by a fifth element in order to form 3-barycentric-free orbits in X105 . ThenBO(3, Z10)≤5.
Lemma 3.11. BO(3, Z4) = 3.
Proof. Lemma 2.6.
Lemma 3.12. BO(4, Z7) = 5.
Proof. Under the action ofG7, the only two orbits inX74areθ({0,1,2,3}) (non barycentric, thus 5 ≤ BO(4, Z7)) and θ({0,1,2,4}) (barycentric). By a simple inspection we can see that the only orbit in X75, obtained from θ({0,1,2,3}), contains a 4-barycentric set. Then BO(4, Z7) ≤ 5. By Lem- mas 2.3 and 2.6, X75 contains only one orbit, say θ({0,1,2,3,4}). Moreover, this orbit contains a 4-barycentric set. Therefore, this is another way to obtain the upper bound BO(4, Z7)≤5.
The two theorems below and their corollaries give other bounds for BO(k, Zp).
Theorem 3.13 ([6]). Let s, d be integers ≥ 2 such that s > lp−1
d
m . Let A be a set with |A| = s+d. Let f : A → Zp be a sequence such that
|f(A)| ≥lp−1
d
m
+d. Then there exists an s-barycentric subsequence of f.
Corollary 3.14. BO lp−1
d
m
+ 1, Zp
≤ l
p−1 d
m
+ 1 +d for d≥ 2 and p≥l
p−1 d
m
+ 1 +d.
Theorem3.15 ([6]). Let s≥2,p≥7 andAa set with s+ 2elements. If f :A→Zp is a sequence with|f(A)|= p+32 , then f contains an s-barycentric subsequence.
Corollary 3.16. BO lp−1
2
m , Zp
≤ p+32 for p≥7.
In particular, we haveBO(5, Z11)≤7.
We have the following easy
Lemma 3.17. θ({0,1,2}) is a barycentric orbit of Xn3, n≥3.
θ({0,1,2,5}) is a barycentric orbit of Xn4,n≥6.
θ({0,1,2,3,4}) is a barycentric orbit of Xn5, n≥5.
θ({0,1,2,3,5,7}) is a barycentric orbit of Xn6, n≥8.
θ({0,1,2,3,4,5,6}) is a barycentric orbit of Xn7, n≥7.
θ({0,1,2,3,4,5,6,7}) is a barycentric orbit of Xn8, n= 10,12.
θ({0,1,2,3,4,7,8,9}) is a barycentric orbit of X118 .
θ({0,1,2,3,4,5,6,7,8}) is a barycentric orbit of Xn9, n≥9.
θ({0,1,2,3,4,5,6,8,10,11}) is a barycentric orbit of X1210.
θ({0,1,2,3,4,5,6,7,8,9,10}) is a barycentric orbit of Xn11, n≥11.
Remark 3.18. The existence ofBO(k, Zn) for 3≤n≤12 and 3≤k≤n is ensured by Remark 3.2 and Lemma 3.17.
The exact values of the k-barycentric Olson constant are presented in Table 1 and Table 2. The values that are not justified by lemmas are obtained similarly using orbits technique as in Lemmas 3.9 through 3.12. In the lower bound column, a longest set without a k-barycentric set is given.
TABLE 1 Exact values ofBO(k, G)
Lower bound Justification k G BO(k, G)
{0,1} 3 Z3 3
{0,1} Lemma 3.11 3 Z4 3
{0,1} 3 Z5 3
{0,1,3,4} Lemma 3.10 3 Z6 5
{0,1,3} 3 Z7 4
{0,1,3,4} Lemma 3.10 3 Z8 5
{0,1,3,4} 3 Z9 5
{0,1,3,4} Lemma 3.10 3 Z10 5
{0,1,4,5} 3 Z11 5
{0,1,3,4} 3 Z12 5
Remark 3.2 4 Z4 non defined Remark 3.2 4 Z5 non defined
{0,1,3,4} 4 Z6 5
{0,1,2,3} Lemma 3.12 4 Z7 5
{0,1,2,5,7} 4 Z8 6
{0,1,3,4,6,7} 4 Z9 7
{0,1,2,3,4} 4 Z10 6
{0,1,2,3,4} 4 Z11 6
{0,1,2,3,4} 4 Z12 6
Remark 3.2 5 Z5 5
{0,1,2,3} 5 Z6 5
{0,1,2,4} 5 Z7 5
{0,1,2,3,5} 5 Z8 6
{0,1,2,3,5} 5 Z9 6
{0,1,2,3,5,8} 5 Z10 7 {0,1,2,6,8,9} 5 Z11 7 {0,1,2,3,5,7} 5 Z12 7
TABLE 2 Exact values ofBO(k, G)
Lower bound Justification k G BO(k, G) Remark 3.2 6 Z6 non defined Remark 3.2 6 Z7 non defined
{0,1,2,3,4,5} 6 Z8 7
{0,1,2,3,4,6} 6 Z9 8
{0,1,2,3,4,5} 6 Z10 7
{0,1,2,3,5,10} 6 Z11 7
{0,1,2,3,5,8} 6 Z12 7
{0,1,2,3,4,5} Remark 3.2 7 Z7 7
{0,1,2,3,4,5} 7 Z8 7
{0,1,2,3,4,5} 7 Z9 7
{0,1,2,3,4,5,7} 7 Z10 8
{0,1,2,3,5,7,10} 7 Z11 8 {0,1,3,4,5,7,10} 7 Z12 8
Remark 3.2 8 Z8 non defined Remark 3.2 8 Z9 non defined {0,1,2,3,4,5,6,8} 8 Z10 9 {0,1,2,3,5,7,9,10} 8 Z11 9 {0,1,2,3,4,5,7,8} 8 Z12 9 {0,1,2,3,4,6,7,8} Remark 3.2 9 Z9 9 {0,1,2,3,5,6,7,8} 9 Z10 9 {0,1,2,3,4,5,6,7} 9 Z11 9 {0,1,2,3,4,5,6,7} 9 Z12 9
Remark 3.2 10 Z10 non defined Remark 3.2 10 Z11 non defined {0,1,2,3,4,5,6,7,8,9} 10 Z12 11 {0,1,2,3,4,5,6,7,8,10} Remark 3.2 11 Z11 11 {0,1,2,3,4,5,6,7,8,9} 11 Z12 11
Remark 3.2 12 Z12 non defined
4. k-BARYCENTRIC DAVENPORT CONSTANT
AND BARYCENTRIC RAMSEY NUMBERS FOR STARS
Let G be an Abelian group of order n and k ≤ n. If BO(k, G) exists, then we have BO(k, G)≤BD(k, G), otherwise n+ 1≤BD(k, G).From the Hamidoune-condition we haveBD(k, G)≤n+k−1, i.e., so that its existence is assured. When k = |G| = n, the k-barycentric sequences are zero-sum.
Gao [9] showed that ZS(G) = n+D(G)−1, where ZS(G) is the smallest positive integer t such that every sequence of length t contains a zero-sum n-subsequence. Then BD(n, G) = ZS(G). Moreover, since D(Zn) = n (see [17]), we have BD(n, G) = 2n−1.
New values of BD(k, G) different from those presented in [7] are given here.
In this section, the values ofBR(K1,k, Zn) for 3≤n≤12 and 3≤k≤n are established. The upper bound is derived from the fact thatBR(K1,k, Zn)≤ BD(k, Zn) + 1. The process of obtaining the lower bound is as follows. Letµ be the sequence that yields the lower bound ofBD(k, Zn).Select an adequate decomposition of the complete graph KBD(k,Zn), using Theorem 2.9, Corol- lary 2.10 and Corollary 2.11. The edges of each subgraph, part of this decom- position, are colored;µis the sequence of edge colors of theK1,BD(k,Zn)−1stars.
In some cases, such asBR(K1,4, Z6) = 7, its upper bound is derived from BO(4, Z6) = 5: if there exists a K1,6 inK7 colored with five or six colors, we are done. Else, if allK1,6 are colored with three or four colors, then each orbit of X63, X64, respectively, is used to color the edges of K7. For example, let θ({0,1,2}) be an orbit of X63. It is easy to see that coloring any K1,6 in K7
with 0,1,2 we can identify aK1,4 barycentric. If allK1,6are colored with only two colors; then they must be colored by {0,1}, {0,2} or {0,3}. Therefore, assuming that K7 is K1,4 barycentric free, the graph induced by the edges colored by 0 has odd order and each vertex has odd degree, and we reach a contradiction. For the lower bound, Theorem 2.9 is used. The two-edge disjoint Hamiltonian cycles of K6are colored by 1 and 3, respectively, and the perfect matching by 0.
In what follows we give a few representative examples to illustrate the methods used to compute BD(k, G).
Lemma 4.1. BD(3, Z4) = 5.
Proof. Since BO(3, Z4) = 3, the sequence that yields the lower bound must have at most two different elements. Since the orbits of X42 areθ({0,1}) and θ({0,2}), the length of 3-barycentric free sequence 0011 is a lower bound for BD(3, Z4). Moreover, in every sequence of length 5, there exist at least three different elements or three identical elements. Then it contains a 3- barycentric sequence. The proof is complete.
Lemma 4.2. BD(3, Z6) = 5.
Proof. Since BO(3, Z6) = 5, the lower bound is obtained for sequences with four different elements. For example, the sequence 0134 is 3-barycentric free. Moreover, if in a sequence of length 5 there exists an element repeated three times, then we are done. Otherwise, it contains three different elements.
Since the only orbit inX63 not barycentric isθ({0,1,3}), it is sufficient to con- sider the sequence 00113, which contains the 3-barycentric sequence 003.
In Table 3 and Table 4 we give the exact values of BD(k, Zn) and BR(K1,k, Zn) for 3 ≤ n ≤ 12 and 3 ≤ k ≤ n. In the lower bound column is given a longest sequence without a k-barycentric subsequence.
TABLE 3
Exact values ofBD(k, G) andBR(K1,k, G) Lower bound k G BD(k, G) BR(K1,k, G)
0101 3 Z3 5 6
0101 3 Z4 5 6
0101 3 Z5 5 6
0113 3 Z6 5 6
013013 3 Z7 7 8
013013 3 Z8 7 8
01340134 3 Z9 9 10
01340134 3 Z10 9 10
01340134 3 Z11 9 10
01340134 3 Z12 9 10
000111 4 Z4 7 7
000111 4 Z5 7 8
000111 4 Z6 7 7
0001222 4 Z7 8 9
01450011 4 Z8 8 9
0120022 4 Z9 8 9
01560011 4 Z10 9 10
01341133 4 Z11 9 10
01270022 4 Z12 9 10
00001111 5 Z5 9 10
00001111 5 Z6 9 10
00001111 5 Z7 9 10
00001111 5 Z8 9 10
00001111 5 Z9 9 10
00001111 5 Z10 9 10
00001111 5 Z11 9 10
000011115 5 Z12 10 11
5. CONCLUSION
The k-barycentric Olson constants BO(k, G) play an important role in the computation of other constants such as the k-barycentric Davenport con- stant BD(k, G) and the barycentric Ramsey numbers starsBR(K1,k, G). We only discussed the existence ofBO(k, G) for some groupG. It is an open prob- lem to prove the existence of and to computeBO(k, G) for a general Abelian group Gby using results of additive group theory.
TABLE 4
Exact values ofBD(k, G) andBR(K1,k, G)
Lower bound k G BD(k, G) BR(K1,k, G)
0000011111 6 Z6 11 11
0000011111 6 Z7 11 12
0000011111 6 Z8 11 11
0000011111 6 Z9 11 12
0000011111 6 Z10 11 11
00000122222 6 Z11 12 13
000001111167 6 Z12 13 14
000000111111 7 Z7 13 14
000000111111 7 Z8 13 14
000000111111 7 Z9 13 14
000000111111 7 Z10 13 14
000000111111 7 Z11 13 14
000000111111 7 Z12 13 14
00000001111111 8 Z8 15 15
00000001111111 8 Z9 15 16
00000001111111 8 Z10 15 15
00000001111111 8 Z11 15 16
00000001111111 8 Z12 15 15
0000000011111111 9 Z9 17 18
0000000011111111 9 Z10 17 18
0000000011111111 9 Z11 17 18
0000000011111111 9 Z12 17 18
000000000111111111 10 Z10 19 19
000000000111111111 10 Z11 19 20
000000000111111111 10 Z12 19 19
00000000001111111111 11 Z11 21 22 00000000001111111111 11 Z12 21 22 0000000000011111111111 12 Z12 23 23
REFERENCES
[1] A. Bialostocki and P. Dierker,On the Erd˝os, Ginzburg and Ziv theorem and the Ramsey numbers for stars and matchings. Discrete Math.1101992, 1–8.
[2] Y. Caro,Zero-sum problems: a survey. Discrete Math.152(1996), 93–113.
[3] Y. Caro, A complete characterization of the zero-sum (mod 2) Ramsey numbers. J.
Combin. TheoryA 68(1994), 205–211.
[4] Y. Caro,On zero-sum Ramsey numbers–stars. Discrete Math.104(1992), 1–6.
[5] L. Childs,Introduction to Higher Algebra. Springer-Verlag, 1979.
[6] C. Delorme, I. M´arquez, O. Ordaz and A. Ortu˜no, Existence condition for barycentric sequences. Discrete Math.281(2004), 163–172.
[7] C. Delorme, S. Gonz´alez, O. Ordaz and M.T. VarelaBarycentric sequences and barycen- tric Ramsey numbers stars. Discrete Math.277(2004), 45–56.
[8] J.A. Dias da Silva and Y.O. Hamidoune,Cyclic spaces for Grassmann derivatives and additive theory. Bull. London Math. Soc.26(1994), 140–146.
[9] W. Gao, Some problems in additive group theory and additive number theory. Ph.D.
Dissertation, 1994.
[10] D.J. Grynkiewicz, A weighted Erd˝os-Ginzburg-Ziv theorem. Combinatorica 26(2006), 445–453.
[11] Y.O. Hamidoune, On weighted sums in Abelian groups. Discrete Math. 162 (1996), 127–132.
[12] Y.O. Hamidoune, Adding distinct congruence classes. Combin. Probab. Comput. 7 (1998), 81–87.
[13] Y.O. Hamidoune, Additive group theory applied to network topology. In: Ding-Zhu Du and D. Frank Hsu (Eds.),Combinatorial Network Theory, pp. 1–39. Kluwer, Dordrecht, 1996.
[14] John B. Fraleigh,A First Course in Abstract Algebra. Addison Wesley, Reading, Mass., 1967.
[15] F. Harary,Graph Theory. Addison Wesley, Reading, Mass., 1969.
[16] Serge Lang,Algebra. Addison Wesley, 1965.
[17] J.E. Olson, A combinatorial problem on finite Abelian groups, I. J. Number Theory1 (1969), 195–199.
Received 2 November 2007 Universidad Central de Venezuela Facultad de Ciencias
Departamento de Matem´aticas y Centro ISYS Ap. 47567, Caracas 1041-A, Venezuela
flosav@cantv.net Universidad Sim´on Bolivar
Departamento de Matem´aticas Puras y Aplicadas Ap. 89000, Caracas 1080-A, Venezuela
and
Universidad de Oriente Departamento de Matem´aticas Escuela de Ciencias, N´ucleo Sucre
Cuman´a, Venezuela
