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1.
A cycle of C of a graph G is called a Dλ-cycle if every component of G ? V(C) has order less than λ. A Dλ-path is defined analogously. In particular, a D1-cycle is a hamiltonian cycle and a D1-path is a hamiltonian path. Necessary conditions and sufficient conditions are derived for graphs to have a Dλ-cycle or Dλ-path. The results are generalizations of theorems in hamiltonian graph theory. Extensions of notions such as vertex degree and adjacency of vertices to subgraphs of order greater than 1 arise in a natural way.  相似文献   

2.
The crossing number of the Cartesian product C3 × Cn of a 3-cycle and an n-cycle is shown to be n.  相似文献   

3.
A k-cycle decomposition of a complete multipartite graph is said to be gregarious if each k-cycle in the decomposition has its vertices in k different partite sets. Equipartite gregarious 3-cycle systems are 3-GDDs, and necessary and sufficient conditions for their existence are known (see for instance the CRC Handbook of Combinatorial Designs, 1996, C.J. Colbourn, J.H. Dinitz (Eds.), Section III 1.3). The cases of equipartite and of almost equipartite 4-cycle systems were recently dealt with by Billington and Hoffman. Here, for both 6-cycles and for 8-cycles, we give necessary and sufficient conditions for existence of a gregarious cycle decomposition of the complete equipartite graph Kn(a) (with n parts, n?6 or n?8, of size a).  相似文献   

4.
A dodecagon quadrangle is the graph consisting of two cycles: a 12-cycle (x1,x2,…,x12) and a 4-cycle (x1,x4,x7,x10). A dodecagon quadrangle system of order n and index ρ [ DQS] is a pair (X,H), where X is a finite set of n vertices and H is a collection of edge disjoint dodecagon quadrangles (called blocks) which partitions the edge set of ρKn, with vertex set X. A dodecagon quadrangle system of order n is said to be perfect [PDQS] if the collection of 4-cycles contained in the dodecagon quadrangles form a 4-cycle system of order n and index μ. In this paper we determine completely the spectrum of DQSs of index one and of PDQSs with the inside 4-cycle system of index one.  相似文献   

5.
In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K v . This supplementary solution builds on a result of Raines and Staniszlo.  相似文献   

6.
LetGbe a planar graph with maximum degreeΔ.In this paper,we prove that if any4-cycle is not adjacent to ani-cycle for anyi∈{3,4}in G,then the list edge chromatic numberχl(G)=Δand the list total chromatic numberχl(G)=Δ+1.  相似文献   

7.
On an algebraic varietyY ? ? N we will call complete intersection a 0-cycle when it is the intersection of Y with a codimension n complete intersection of ? N . We consider the following problem: Let E?Y be given. Does E contain the support of a complete intersection 0-cycle? The two main theorems shown in this article give the answers in some cases: first, a negative answer for E some “big” subset of a singular irreducible algebraic variety; secondly, a positive answer for some “small” subset, on any algebraic variety.  相似文献   

8.
A V-cycle multigrid method is developed for a time-dependent viscoelastic fluid flow satisfying an Oldroyd-B-type constitutive equation in two-dimensional domains. Also existence, uniqueness, and error estimates of an approximate solution are discussed. The approximate stress, velocity, and pressure are, respectively, σ k -discontinuous, u k -continuous, and p k -continuous.  相似文献   

9.
In this part II, I study the class C of an n-cycle or of an n ? 1-cycle in the alternating group An. For n = 4k ? 1, 4k, CCC covers An, but CC does not. For n = 4k + 1, CCC covers An; I do not know whether CC does. For n = 4k + 2, CC covers An [Part III concerns PSL(2, q).]  相似文献   

10.
For a smooth projective variety X of dimension n, on the product of Chow varieties Ca(XCna−1(X) parameterizing pairs (A,B) of an a-cycle A and an (na−1)-cycle B in X, Barry Mazur raised the problem of constructing a Cartier divisor supported on the locus of pairs with AB≠0?. We introduce a new approach to this problem, and new techniques supporting this approach.  相似文献   

11.
Chin-Mei Fu 《Discrete Mathematics》2008,308(13):2901-2909
Let G be the set that contains precisely the graphs on n vertices with maximum degree 3 for which there exists a 4-cycle system of their complement in Kn. In this paper G is completely characterized.  相似文献   

12.
In real n-space the orthant monotonic norms of Gries [5] can be given a new characterization similar to one for monotonic norms: a norm is orthant monotonic if and only if for every D=diag(δ1,δ2,…,δn)?0, the operator norm of D equals max δi. This gives an alternative proof to Gries's: a norm is orthant monotonic if and only if its dual norm is orthant monotonic. Also, it follows that the principal axis vectors are self-dual for orthant monotonic norms.  相似文献   

13.
Let G=(V,E) be a graph. A set SV is a restrained dominating set (RDS) if every vertex not in S is adjacent to a vertex in S and to a vertex in V?S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of an RDS of G. A set SV is a total dominating set (TDS) if every vertex in V is adjacent to a vertex in S. The total domination number of a graph G without isolated vertices, denoted by γt(G), is the minimum cardinality of a TDS of G.Let δ and Δ denote the minimum and maximum degrees, respectively, in G. If G is a graph of order n with δ?2, then it is shown that γr(G)?n-Δ, and we characterize the connected graphs with δ?2 achieving this bound that have no 3-cycle as well as those connected graphs with δ?2 that have neither a 3-cycle nor a 5-cycle. Cockayne et al. [Total domination in graphs, Networks 10 (1980) 211-219] showed that if G is a connected graph of order n?3 and Δ?n-2, then γt(G)?n-Δ. We further characterize the connected graphs G of order n?3 with Δ?n-2 that have no 3-cycle and achieve γt(G)=n-Δ.  相似文献   

14.
15.
A class of algebras forms a variety if it is characterised by a collection of identities. There is a well-known method, often called the standard construction, which gives rise to algebras from m-cycle systems. It is known that the algebras arising from {1}-perfect m-cycle systems form a variety for m∈{3,5} only, and that the algebras arising from {1,2}-perfect m-cycle systems form a variety for m∈{3,5,7} only. Here we give, for any set K of positive integers, necessary and sufficient conditions under which the algebras arising from K-perfect m-cycle systems form a variety.  相似文献   

16.
We prove that modular spaces Lρ have the uniform Kadec-Klee property w.r.t. the convergence ρ-a.e. when they are endowed with the Luxemburg norm. We also prove that these spaces have the uniform Opial condition w.r.t. the convergence ρ-a.e. for both the Luxemburg norm and the Amemiya norm. Some assumptions over the modular ρ need to be assumed. The above geometric properties will enable us to obtain some fixed point results in modular spaces for different kind of mappings.  相似文献   

17.
Decompositions of the complete graph with n vertices K n into edge disjoint cycles of length m whose union is K n are commonly called m-cycle systems. Any m-cycle system gives rise to a groupoid defined on the vertex set of K n via a well known construction. Here, it is shown that the groupoids arising from all m-cycle systems are precisely the finite members of a variety (of groupoids) for m = 3 and 5 only.  相似文献   

18.
Let G be a planar graph of maximum degree 6. In this paper we prove that if G does not contain either a 6-cycle, or a 4-cycle with a chord, or a 5- and 6-cycle with a chord, then χ(G)=6, where χ(G) denotes the chromatic index of G.  相似文献   

19.
A block-colouring of a 4-cycle system (V,B) of order v=1+8k is a mapping ?:BC, where C is a set of colours. Every vertex of a 4-cycle system of order v=8k+1 is contained in blocks and r is called, using the graph theoretic terminology, the degree or the repetition number. A partition of degree r into s parts defines a colouring of type s in which the blocks containing a vertex x are coloured exactly with s colours. For a vertex x and for i=1,2,…,s, let Bx,i be the set of all the blocks incident with x and coloured with the ith colour. A colouring of type s is equitable if, for every vertex x, we have |Bx,iBx,j|≤1, for all i,j=1,…,s. In this paper we study bicolourings, tricolourings and quadricolourings, i.e. the equitable colourings of type s with s=2, s=3 and s=4, for 4-cycle systems.  相似文献   

20.
Let ∥·∥ be an operator norm and ∥·∥D its dual. Then it is shown that ∥AD? ∑|λi(A)|, where λi(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.  相似文献   

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