DEGREE CONDITIONS OF INDUCED MATCHING EXTENDABLE GRAPHS

Abstract. A simple graph G is induced matching extendable,shortly IM-extendable,if every in-duced matching of G is included in a perfect matching of G. The degree conditions of IM-extend-able graphs are researched in this paper. The main results are as follows：     

On the order of certain close to regular graphs without a matching of given size

A graph G is a {d, d+k}-graph, if one vertex has degree d+k and the remaining vertices of G have degree d. In the special case of k = 0, the graph G is d-regular. Let k, p ⩾ 0 and d, n ⩾ 1 be integers such that n and p are of the same parity. If G is a connected {d, d+k{-graph of order n without a matching M of size 2|M| = n − p, then we show in this paper the following: If d = 2, then k ⩾ 2(p + 2) and

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

A note on Bol loops of order 2 n k

It is shown that, for eachn 2 and k 3, there exist at least 2 ⁿ -3 non-isomorphic loops of order 2 ⁿ k which are Bol but not Moufang. In most cases this bound can be improved.     

On k -diameter of k -connected graphs

Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d _k (G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d _k (G). For a fixed positive integer d, some conditions to insure d _k (G)⩽d are given in this paper. In particular, if d⩾3 and the sum of degrees of any s (s=2 or 3) nonadjacent vertices is at least n+(s−1)k+1−d, then d _k (G)⩽d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible. Supported by NNSF of China (19971086).     

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献   

A k-Tree Containing Specified Vertices

A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let ${S \subseteq V(G)}A k-tree is a tree with maximum degree at most k. In this paper, we give sufficient conditions for a graph to have a k-tree containing specified vertices. Let k be an integer with k > 3. Let G be a graph of order n and let S í V(G){S \subseteq V(G)} with κ(S) ≥ 1. Suppose that for every l > κ(S), there exists an integer t such that 1 ￡ t ￡ (k-1)l+2 - ?\fracl-1k ?{1 \le t \leq (k-1)l+2 - \lfloor \frac{l-1}{k} \rfloor} and the degree sum of any t independent vertices of S is at least n + tl − kl − 1. Then G has a k-tree containing S. We also show some new results on a spanning k-tree as corollaries of the above theorem.     

Repelling periodic points of given periods of rational functions

Let R(z) be a rational function of degree d≥2. Then R(z) has at least one repelling periodic point of given period k≥2, unless k = 4 and d = 2, or k = 3 and d≤3, or k = 2 and d≤8. Examples show that all exceptional cases occur.     

On extremal sizes of locally k-tree graphs

A graph G is a locally k-tree graph if for any vertex v the subgraph induced by the neighbours of v is a k-tree, k ⩾ 0, where 0-tree is an edgeless graph, 1-tree is a tree. We characterize the minimum-size locally k-trees with n vertices. The minimum-size connected locally k-trees are simply (k + 1)-trees. For k ⩾ 1, we construct locally k-trees which are maximal with respect to the spanning subgraph relation. Consequently, the number of edges in an n-vertex locally k-tree graph is between Ω(n) and O(n ²), where both bounds are asymptotically tight. In contrast, the number of edges in an n-vertex k-tree is always linear in n.     

On Approximate Geometric k -Clustering

For a partition of an n -point set into k subsets (clusters) S ₁ ,S ₂ ,. . .,S _k , we consider the cost function , where c(S _i ) denotes the center of gravity of S _i . For k=2 and for any fixed d and ε >0 , we present a deterministic algorithm that finds a 2-clustering with cost no worse than (1+ε) -times the minimum cost in time O(n log n); the constant of proportionality depends polynomially on ε . For an arbitrary fixed k , we get an O(n log ^k n) algorithm for a fixed ε , again with a polynomial dependence on ε . Received October 19, 1999, and in revised form January 19, 2000.     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

On Hong’s conjecture for power LCM matrices

A set S={x ₁,...,x _n } of n distinct positive integers is said to be gcd-closed if (x _i , x _j ) ∈ S for all 1 ⩽ i, j ⩽ n. Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n ⩽ k(t), then the power LCM matrix ([x _i , x _j ] ^t ) defined on any gcd-closed set S={x ₁,...,x _n } is nonsingular, but for n ⩾ k(t) + 1, there exists a gcd-closed set S={x ₁,...,x _n } such that the power LCM matrix ([x _i , x _j ] ^t ) on S is singular. In 1996, Hong proved k(1) = 7 and noted k(t) ⩾ 7 for all t ⩾ 2. This paper develops Hong’s method and provides a new idea to calculate the determinant of the LCM matrix on a gcd-closed set and proves that k(t) ⩾ 8 for all t ⩾ 2. We further prove that k(t) ⩾ 9 iff a special Diophantine equation, which we call the LCM equation, has no t-th power solution and conjecture that k(t) = 8 for all t ⩾ 2, namely, the LCM equation has t-th power solution for all t ⩾ 2.     

Graph decomposition with applications to subdivisions and path systems modulo k

The existence of a function α(k) (where k is a natural number) is established such that the vertex set of any graph G of minimum degree at least α(k) has a decomposition A ∪ B ∪ C such that G(A) has minimum degree at least k, each vertex of A is joined to at least k vertices of B, and no two vertices of B are separated by fewer than k vertices in G(G ∪ C). This is applied to prove the existence of subdivisions of complete bipartite graphs (complete graphs) with prescribed path lengths modulo k in graphs of sufficiently high minimum degree (chromatic number) and path systems with prescribed ends and prescribed lengths modulo k in graphs of sufficiently high connectivity.     

The Threshold for the Erdos, Jacobson and Lehel Conjecture to Be True

Let σ（k, n） be the smallest even integer such that each n-term positive graphic sequence with term sum at least σ（k, n） can be realized by a graph containing a clique of k ＋ 1 vertices. Erdos et al. （Graph Theory, 1991, 439-449） conjectured that σ（k, n） = （k - 1）（2n- k） ＋ 2. Li et al. （Science in China, 1998, 510-520） proved that the conjecture is true for k 〉 5 and n ≥ （k2） ＋ 3, and raised the problem of determining the smallest integer N（k） such that the conjecture holds for n ≥ N（k）. They also determined the values of N（k） for 2 ≤ k ≤ 7, and proved that [5k-1/2] ≤ N（k） ≤ （k2） ＋ 3 for k ≥ 8. In this paper, we determine the exact values of σ（k, n） for n ≥ 2k＋3 and k ≥ 6. Therefore, the problem of determining σ（k, n） is completely solved. In addition, we prove as a corollary that N（k） -= [5k-1/2] for k ≥6.     

Isometric embeddings of Johnson graphs in Grassmann graphs

Let V be an n-dimensional vector space (4≤n<∞) and let G_k(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 G_k(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.     

On Enomoto’s problems in a bipartite graph

In this paper, we obtain the following result: Let k, n ₁ and n ₂ be three positive integers, and let G = (V ₁,V ₂;E) be a bipartite graph with |V1| = n ₁ and |V ₂| = n ₂ such that n ₁ ⩾ 2k + 1, n ₂ ⩾ 2k + 1 and |n ₁ − n ₂| ⩽ 1. If d(x) + d(y) ⩾ 2k + 2 for every x ∈ V ₁ and y ∈ V ₂ with xy $ \notin $ \notin E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Degree Bounded Spanning Trees

In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set ${S \subseteq V(G)}In this paper, we give a sufficient condition for a graph to have a degree bounded spanning tree. Let n ≥ 1, k ≥ 3, c ≥ 0 and G be an n-connected graph. Suppose that for every independent set S í V(G){S \subseteq V(G)} of cardinality n(k−1) + c + 2, there exists a vertex set X í S{X \subseteq S} of cardinality k such that the degree sum of vertices in X is at least |V(G)| − c −1. Then G has a spanning tree T with maximum degree at most k+éc/nù{k+\lceil c/n\rceil} and ?_{v ? V(T)}max{d_T(v)-k,0} ￡ c{\sum_{v\in V(T)}\max\{d_T(v)-k,0\}\leq c} .     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.