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1.
Let G = (V, E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A ⊆ V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We give an efficient algorithm for finding a maximum collection
of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem.
This research was partially conducted during the period Chudnovsky served as a Clay Mathematics Institute Long-Term Prize
Fellow. The research was supported in part by the Natural Sciences and Engineering Council of Canada. 相似文献
2.
Paul Wollan 《Combinatorica》2011,31(1):95-126
We prove that for all positive integers k, there exists an integer N =N(k) such that the following holds. Let G be a graph and let Γ an abelian group with no element of order two. Let γ: E(G)→Γ be a function from the edges of G to the elements of Γ. A non-zero cycle is a cycle C such that Σ
e∈E(C)
γ(e) ≠ 0 where 0 is the identity element of Γ. Then G either contains k vertex disjoint non-zero cycles or there exists a set X ⊆ V (G) with |X| ≤N(k) such that G−X contains no non-zero cycle. 相似文献
3.
Large Vertex-Disjoint Cycles in a Bipartite Graph 总被引:4,自引:0,他引:4
Hong Wang 《Graphs and Combinatorics》2000,16(3):359-366
Let s≥2 and k be two positive integers. Let G=(V
1,V
2;E) be a bipartite graph with |V
1|=|V
2|=n≥s
k and the minimum degree at least (s−1)k+1. When s=2 and n >2k, it is proved in [5] that G contains k vertex-disjoint cycles. In this paper, we show that if s≥3, then G contains k vertex-disjoint cycles of length at least 2s.
Received: March 2, 1998 Revised: October 26, 1998 相似文献
4.
G. R. T. Hendry 《Periodica Mathematica Hungarica》1990,21(3):205-218
A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply
the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K
k
orK
k
−e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson [13]. 相似文献
5.
Mark Pankov 《Journal of Algebraic Combinatorics》2011,33(4):555-570
Let V be an n-dimensional vector space (4≤n<∞) and let Gk(V){\mathcal{G}}_{k}(V) be the Grassmannian formed by all k-dimensional subspaces of V. The corresponding Grassmann graph will be denoted by Γ
k
(V). We describe all isometric embeddings of Johnson graphs J(l,m), 1<m<l−1 in Γ
k
(V), 1<k<n−1 (Theorem 4). As a consequence, we get the following: the image of every isometric embedding of J(n,k) in Γ
k
(V) is an apartment of Gk(V){\mathcal{G}}_{k}(V) if and only if n=2k. Our second result (Theorem 5) is a classification of rigid isometric embeddings of Johnson graphs in Γ
k
(V), 1<k<n−1. 相似文献
6.
Peter Abramenko 《Israel Journal of Mathematics》1994,87(1-3):203-223
LetG be a simple Chevalley group of rankn and Γ=G(
). Then the finiteness length of Γ shall be determined by studying the action of Γ on the Bruhat-Tits buildingX ofG
. This is always possible provided that certain subcomplexes of the links of simplices inX are spherical. As a consequence, one obtains that Γ is of typeF
n−1 but not of typeFP
n ifG is of typeA
n, Bn, Cn orD
n andq≥22n−1. 相似文献
7.
Bin Yong SUN 《数学学报(英文版)》2008,24(2):305-310
The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G. 相似文献
8.
Dr. Matthias Kriesell 《Combinatorica》2006,26(3):277-314
A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism).
Here we prove this conjecture. 相似文献
9.
Thek-Centrum Shortest Path Problem (kCSP[s, t]) is to minimize the sum of thek longest arcs in any (simple)s−t path containing at leastk arcs, wherek is a positive integer.kCSP is introduced and is shown to be NP-Hard, although it is polynomially solvable ifk is constrained to be no greater than the number of arcs in ans−t path with fewest arcs. Some properties of the problem are studied and a new weakly dual problem is also introduced. 相似文献
10.
Hong Wang 《Graphs and Combinatorics》2001,17(1):177-183
Let G=(V
1,V
2;E) be a bipartite graph with 2k≤m=|V
1|≤|V
2|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs.
Received: December 9, 1998 Final version received: June 2, 1999 相似文献
11.
C S Rajan 《Proceedings Mathematical Sciences》1994,104(2):389-395
LetG be a connected complex semisimple Lie group. Let Γ be a cocompact lattice inG. In this paper, we show that whenG isSL
2(C), nontrivial deformations of the canonical complex structure onX exist if and only if the first Betti number of the lattice Γ is non-zero. It may be remarked that for a wide class of arithmetic
groups Γ, one can find a subgroup Γ′ of finite index in Γ, such that Γ′/[Γ′,Γ′] is finite (it is a conjecture of Thurston
that this is true for all cocompact lattices inSL(2, C)).
We also show thatG acts trivially on the coherent cohomology groupsH
i(Γ/G, O) for anyi≥0. 相似文献
12.
For a graph G, we define σ2(G) := min{d(u) + d(v)|u, v ≠ ∈ E(G), u ≠ v}. Let k ≥ 1 be an integer and G be a graph of order n ≥ 3k. We prove if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v
1,...,v
k
, G has k vertex-disjoint cycles C
1,..., C
k
of length at most four such that v
i
∈ V(C
i
) for all 1 ≤ i ≤ k. And show if σ2(G) ≥ n + k − 1, then for any set of k independent vertices v
1,...,v
k
, G has k vertex-disjoint cycles C
1,..., C
k
such that v
i
∈ V(C
i
) for all 1 ≤ i ≤ k, V(C
1) ∪...∪ V(C
k
) = V(G), and |C
i
| ≤ 4 for all 1 ≤ i ≤ k − 1.
The condition of degree sum σ2(G) ≥ n + k − 1 is sharp.
Received: December 20, 2006. Final version received: December 12, 2007. 相似文献
13.
A path cover of a graph G=(V,E) is a family of vertex-disjoint paths that covers all vertices in V. Given a graph G, the path cover problem is to find a path cover of minimum cardinality. This paper presents a simple O(n)-time approximation algorithm for the path cover problem on circular-arc graphs given a set of n arcs with endpoints sorted. The cardinality of the path cover found by the approximation algorithm is at most one more than the optimal one. By using the result, we reduce the path cover problem on circular-arc graphs to the Hamiltonian cycle and Hamiltonian path problems on the same class of graphs in O(n) time. Hence the complexity of the path cover problem on circular-arc graphs is the same as those of the Hamiltonian cycle and Hamiltonian path problems on circular-arc graphs. 相似文献
14.
Hatem Hamrouni 《manuscripta mathematica》2008,127(4):511-519
Let G be a connected and simply connected nilpotent Lie group and A a closed connected subgroup of G. Let Γ be a discrete cocompact subgroup of G. In the first part of this paper we give the direct integral decomposition of the up–down representation . As a consequence, we establish a necessary and sufficient condition for A to act ergodically on G/Γ in the case when Γ is a lattice subgroup of G and A is a one-parameter subgroup of G. 相似文献
15.
Dan Yasaki 《Selecta Mathematica, New Series》2006,12(3-4):541-564
Let X = Γ \G/ K be an arithmetic quotient of a symmetric space of non-compact type. In the case that G has
-rank 1, we construct Γ-equivariant deformation retractions of D = G/K onto a set D0. We prove that D0 is a spine, having dimension equal to the virtual cohomological dimension of Γ. In fact, there is a (k − 1)-parameter family of such deformation retractions, where k is the number of Γ -conjugacy classes of rational parabolic subgroups of G. The construction of the spine also gives a way to construct an exact fundamental domain for Γ. 相似文献
16.
Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H\ G. For an increasing family of finite subsets {Γ
T
: T > 0}, a dense orbit υ· Γ, υ∈V and compactly supported function φ on V, we consider the sums
. Understanding the asymptotic behavior of S
φ,υ
(T) is a delicate problem which has only been considered for certain very special choices of H,G and {Γ
T
}. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have
where G
T
= {g ∈G: ||g|| < T} and Γ
T
= G
T
∩ Γ. We also show that the asymptotics of S
φ, υ
(T) is governed by
where ν is an explicit limiting density depending on the choice of υ and || · ||.
Submitted: March 2005 Revision: April 2006 Accepted: June 2006 相似文献
17.
In this paper we consider the k-fixed-endpoint path cover problem on proper interval graphs, which is a generalization of the path cover problem. Given a
graph G and a set T of k vertices, a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint simple paths that covers the vertices of G, such that the vertices of T are all endpoints of these paths. The goal is to compute a k-fixed-endpoint path cover of G with minimum cardinality. We propose an optimal algorithm for this problem with runtime O(n), where n is the number of intervals in G. This algorithm is based on the Stair Normal Interval Representation (SNIR) matrix that characterizes proper interval graphs. In this characterization, every maximal clique of the graph is represented by
one matrix element; the proposed algorithm uses this structural property, in order to determine directly the paths in an optimal
solution. 相似文献
18.
Let k ≥ 2 be an integer. We show that if G is a (k + 1)-connected graph and each pair of nonadjacent vertices in G has degree sum at least |G| + 1, then for each subset S of V(G) with |S| = k, G has a spanning tree such that S is the set of endvertices. This result generalizes Ore’s theorem which guarantees the existence of a Hamilton path connecting
any two vertices.
Dedicated to Professor Hikoe Enomoto on his 60th birthday. 相似文献
19.
Heinz Gröflin 《Combinatorica》1984,4(4):281-290
Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR
v
, using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the
problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s
theorem. 相似文献
20.
In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of
subsets {A
t
| t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements f
t
of G under which #{t∈ℕ | f
t
x∈A
t
} is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical
Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.
Oblatum 27-VII-1998 & 2-IV-1999 / Published online: 5 August 1999 相似文献