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1.
. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T
i
}, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏
i
=1
n
T
i
admits an (r, d)-invariant orientation provided that d(T
1)≥d(T
2)≥4 for n=2, and d(T
1)≥5 and d(T
2)≥4 for n≥3.
Received: July 30, 1997 Final version received: April 20, 1998 相似文献
2.
A finite group G is called p
i
-central of height k if every element of order p
i
of G is contained in the k
th
-term ζ
k
(G) of the ascending central series of G. If p is odd, such a group has to be p-nilpotent (Thm. A). Finite p-central p-groups of height p − 2 can be seen as the dual analogue of finite potent p-groups, i.e., for such a finite p-group P the group P/Ω1(P) is also p-central of height p − 2 (Thm. B). In such a group P, the index of P
p
is less than or equal to the order of the subgroup Ω1(P) (Thm. C). If the Sylow p-subgroup P of a finite group G is p-central of height p − 1, p odd, and N
G
(P) is p-nilpotent, then G is also p-nilpotent (Thm. D). Moreover, if G is a p-soluble finite group, p odd, and P ∈ Syl
p
(G) is p-central of height p − 2, then N
G
(P) controls p-fusion in G (Thm. E). It is well-known that the last two properties hold for Swan groups (see [11]). 相似文献
3.
S. A. Shakhova 《Algebra and Logic》2006,45(4):277-285
Let ℳ be any quasivariety of Abelian groups, Lq(ℳ) be a subquasivariety lattice of ℳ, dom
G
ℳ
be the dominion of a subgroup H of a group G in ℳ, and G/dom
G
ℳ
(H) be a finitely generated group. It is known that the set L(G, H, ℳ) = {dom
G
N
(H)| N ∈ Lq(ℳ)} forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of dom
G
ℳ
(H). It is proved that the lattice L(G,H,ℳ) is semidistributive and necessary and sufficient conditions are specified for
its being distributive.
__________
Translated from Algebra i Logika, Vol. 45, No. 4, pp. 484–499, July–August, 2006. 相似文献
4.
Gustavo A. Fernández-Alcober 《Israel Journal of Mathematics》2007,162(1):75-79
Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all i ≥ 0: (i) exp Ωi(G) ≤ p
i for odd p, and expΩi(G) ≤ 2
i+1 for p = 2; (ii) the index |G: G
p
i| coincides with the number of elements of G of order at most p
i.
Supported by the Spanish Ministry of Science and Education, grant MTM2004-04665, partly with FEDER funds, and by the University
of the Basque Country, grant UPV05/99. 相似文献
5.
A. Schrijver 《Discrete and Computational Geometry》1991,6(1):527-574
In this paper we describe a polynomial-time algorithm for the following problem:given: a planar graphG embedded in ℝ2, a subset {I
1, …,I
p} of the faces ofG, and pathsC
1, …,C
k inG, with endpoints on the boundary ofI
1 ∪ … ∪I
p; find: pairwise disjoint simple pathsP
1, …,P
k inG so that, for eachi=1, …,k, P
i is homotopic toC
i in the space ℝ2\(I
1 ∪ … ∪I
p).
Moreover, we prove a theorem characterizing the existence of a solution to this problem. Finally, we extend the algorithm
to disjoint homotopic trees. As a corollary we derive that, for each fixedp, there exists a polynormial-time algorithm for the problem:given: a planar graphG embedded in ℝ2 and pairwise disjoint setsW
1, …,W
k of vertices, which can be covered by the boundaries of at mostp faces ofG;find: pairwise vertex-disjoint subtreesT
1, …,T
k ofG whereT
i
(i=1, …, k). 相似文献
6.
7.
Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ
p(X) as inf{Σ
i
=1/m
|x*(x
i)|p
p Σ
i
=1/m
‖x
i‖p
p]1
p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x
1,x
2, …,x
m} ⊂X such that Σ
i
=1/m
‖x
i‖>0. It follows immediately from [2] thatμ
p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ
p(X) for various spaces, and obtain some asymptotic estimates ofμ
p(X) for general finite dimensional Banach spaces.
This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof.
A. Dvoretzky and Prof. J. Lindenstrauss. 相似文献
8.
Let G be a finite group. A subgroup K of a group G is called an ?-subgroup of G if N G (K) ∩ K x ≦ K for all x ? G. The set of all ?-subgroups of G will be denoted by ?(G). Let P be a nontrivial p-group. A chain of subgroups 1 = P 0 ? P 1 ? ··· ? P n = P is called a maximal chain of P provided that |P i : P i?1| = p, i = 1, 2, ···, n. A nontrivial p-subgroup P of G is called weakly supersolvably embedded in G if P has a maximal chain 1 = P 0 ? P 1 ? ··· ? P i ? ··· ? P n = P such that P i ? ?(G) for i = 1, 2, ···, n. Using the concept of weakly supersolvably embedded, we obtain new characterizations of p-nilpotent and supersolvable finite groups. 相似文献
9.
Let G be a finite group. A subgroup H of G is called a CAP-subgroup if the following condition is satisfied: for each chief factor K/L of G either HK = HL or H ∩ K = H ∩ L. Let p be a prime factor of |G| and let P be a Sylow p-subgroup of G. If d is the minimum number of generators of P then there exists a family of maximal subgroups of P, denoted by M d (P)={P 1, P 2,…, P d } such that ∩ i=1 d P i = ?(P). In this paper, we investigate the group G satisfying the condition: every member of a fixed M d (P) is a CAP-subgroup of G. For example, if, in addition, G is p-solvable, then G is p-supersolvable. 相似文献
10.
Let G be a graph of order n with connectivity κ≥3 and let α be the independence number of G. Set σ4(G)= min{∑4
i
=1
d(x
i
):{x
1,x
2,x
3,x
4} is an independent set of G}. In this paper, we will prove that if σ4(G)≥n+2κ, then there exists a longest cycle C of G such that V(G−C) is an independent set of G. Furthermore, if the minimum degree of G is at least α, then G is hamiltonian.
Received: July 31, 1998?Final version received: October 4, 2000 相似文献
11.
Daniel C. Mayer 《Monatshefte für Mathematik》2012,179(1):467-495
Explicit expressions for the transfers V i from a metabelian p-group G of coclass cc(G) = 1 to its maximal normal subgroups M 1, . . . , M p+1 are derived by means of relations for generators. The expressions for the exceptional case p = 2 differ significantly from the standard case of odd primes p ≥ 3. In both cases the transfer kernels Ker(V i ) are calculated and the principalisation type of the metabelian p-group is determined, if G is realised as the Galois group Gal(Fp2(K)|K){{\rm{Gal}}({F}_p^2(K)\vert K)} of the second Hilbert p-class field Fp2(K){{F}_p^2(K)} of an algebraic number field K. For certain metabelian 3-groups G with abelianisation G/G′ of type (3, 3) and of coclass cc(G) = r ≥ 3, it is shown that the principalisation type determines the position of G on the coclass graph G(3,r){\mathcal{G}(3,r)} in the sense of Eick and Leedham-Green. 相似文献
12.
Yakov Berkovich 《Israel Journal of Mathematics》2010,180(1):371-412
Let G be a finite p-group. If p = 2, then a nonabelian group G = Ω1(G) is generated by dihedral subgroups of order 8. If p > 2 and a nonabelian group G = Ω1(G) has no subgroup isomorphic to Sp2{\Sigma _{{p^2}}}, a Sylow p-subgroup of the symmetric group of degree p
2, then it is generated by nonabelian subgroups of order p
3 and exponent p. If p > 2 and the irregular p-group G has < p nonabelian subgroups of order p
p
and exponent p, then G is of maximal class and order p
p+1. We also study in some detail the p-groups, containing exactly p nonabelian subgroups of order p
p
and exponent p. In conclusion, we prove three new counting theorems on the number of subgroups of maximal class of certain type in a p-group. In particular, we prove that if p > 2, and G is a p-group of order > p
p+1, then the number of subgroups ≅ ΣSp2{\Sigma _{{p^2}}} in G is a multiple of p. 相似文献
13.
Let π = (d
1, d
2, ..., d
n
) and π′ = (d′
1, d′
2, ..., d′
n
) be two non-increasing degree sequences. We say π is majorizated by π′, denoted by π ⊲ π′, if and only if π ≠ π′, Σ
i=1
n
d
i
= Σ
i=1
n
d′
i
, and Σ
i=1
j
d
i
≤ Σ
i=1
j
d′
i
for all j = 1, 2, ..., n. Weuse C
π
to denote the class of connected graphs with degree sequence π. Let ρ(G) be the spectral radius, i.e., the largest eigenvalue of the adjacent matrix of G. In this paper, we extend the main results of [Liu, M. H., Liu, B. L., You, Z. F.: The majorization theorem of connected
graphs. Linear Algebra Appl., 431(1), 553–557 (2009)] and [Bıyıkoğlu, T., Leydold, J.: Graphs with given degree sequence and maximal spectral radius. Electron. J. Combin., 15(1), R119 (2008)]. Moreover, we prove that if π and π′ are two different non-increasing degree sequences of unicyclic graphs with π ⊲ π′, G and G′ are the unicyclic graphs with the greatest spectral radii in C
π
and C′
π
, respectively, then ρ(G) < ρ(G′). 相似文献
14.
Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ4(G)=min{d(x
1)+d(x
2)+ d(x
3)+d(x
4) | {x
1,…,x
4} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ4(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path.
Received: August 28, 2000 Final version received: May 23, 2002 相似文献
15.
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f: V (G) → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f
1, f
2, …, f
d
} of distinct Roman k-dominating functions on G with the property that Σ
i=1
d
f
i
(v) ≤ 2 for each v ∈ V (G), is called a Roman k-dominating family (of functions) on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by d
kR
(G). Note that the Roman 1-domatic number d
1R
(G) is the usual Roman domatic number d
R
(G). In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for d
kR
(G). In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman
domatic number. 相似文献
16.
OD-characterization of Almost Simple Groups Related to U3(5) 总被引:1,自引:0,他引:1
Let G be a finite group with order |G|=p1^α1p2^α2……pk^αk, where p1 〈 p2 〈……〈 Pk are prime numbers. One of the well-known simple graphs associated with G is the prime graph (or Gruenberg- Kegel graph) denoted .by г(G) (or GK(G)). This graph is constructed as follows: The vertex set of it is π(G) = {p1,p2,…,pk} and two vertices pi, pj with i≠j are adjacent by an edge (and we write pi - pj) if and only if G contains an element of order pipj. The degree deg(pi) of a vertex pj ∈π(G) is the number of edges incident on pi. We define D(G) := (deg(p1), deg(p2),..., deg(pk)), which is called the degree pattern of G. A group G is called k-fold OD-characterizable if there exist exactly k non- isomorphic groups H such that |H| = |G| and D(H) = D(G). Moreover, a 1-fold OD-characterizable group is simply called OD-characterizable. Let L := U3(5) be the projective special unitary group. In this paper, we classify groups with the same order and degree pattern as an almost simple group related to L. In fact, we obtain that L and L.2 are OD-characterizable; L.3 is 3-fold OD-characterizable; L.S3 is 6-fold OD-characterizable. 相似文献
17.
Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω1(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω1(P) ≦ Z(P) and N
G
(Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result.
This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059. 相似文献
18.
J. B. Shearer 《Combinatorica》1985,5(3):241-245
LetX
1, ...,X
n
be events in a probability space. Let ϱi be the probabilityX
i
occurs. Let ϱ be the probability that none of theX
i
occur. LetG be a graph on [n] so that for 1 ≦i≦n X
i
is independent of ≈X
j
‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ1, ..., ϱ
n
≦x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1)
d−1
d
−d ford≧2. Hence
df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ
i
andG. 相似文献
19.
Haruko Okamura 《Graphs and Combinatorics》2005,21(4):503-514
Let k≥2 be an integer and G = (V(G), E(G)) be a k-edge-connected graph. For X⊆V(G), e(X) denotes the number of edges between X and V(G) − X. Let {si, ti}⊆Xi⊆V(G) (i=1,2) and X1∩X2=∅. We here prove that if k is even and e(Xi)≤2k−1 (i=1,2), then there exist paths P1 and P2 such that Pi joins si and ti, V(Pi)⊆Xi (i=1,2) and G − E(P1∪P2) is (k−2)-edge-connected (for odd k, if e(X1)≤2k−2 and e(X2)≤2k−1, then the same result holds [10]), and we give a generalization of this result and some other results about paths not containing
given edges. 相似文献
20.
Haipeng QU 《Frontiers of Mathematics in China》2010,5(1):117-122
For a positive integer n, a finite p-group G is called an ℳ
n
-group, if all subgroups of index p
n
of G are metacyclic, but there is at least one subgroup of index p
n−1 that is not. A classical result in p-group theory is the classification of ℳ1-groups by Blackburn. In this paper, we give a slightly shorter and more elementary proof of this result. 相似文献