共查询到20条相似文献,搜索用时 15 毫秒
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Yoko Usami 《Discrete Mathematics》1983,44(2):195-199
A simple graph with n vertices is called Pi-connected if any two distinct vertices are connected by an elementary path of length i. In this paper, lower bounds of the number of edges in graphs that are both P2- and Pi-connected are obtained. Namely if , then |E(G)|?((4i?5)/(2i?2))(n?1), and if , then |E(G)|?2(n?1) apart from one exeptional graph. Furthermore, extremal graphs are determined in the former. 相似文献
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D.R Woodall 《Journal of Combinatorial Theory, Series B》1973,15(3):225-255
The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) ≥ c, then G contains at least disjoint edges if , at least disjoint edges if , a Hamiltonian circuit if , and a circuit of length at least if and G is not one of two specified exceptional graphs. Each of these results is best possible.The Anderson number of a graph is defined. The Anderson numbers of a few very simple graphs are determined; and some rather weak bounds are obtained, and some conjectures made, on the Anderson numbers of graphs in general. 相似文献
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Let G be any 3-connected graph containing n vertices and r the radius of G. Then the inequality is proved. A similar theorem concerning any (2m ?1)-connected graph G can be proved too. 相似文献
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Simonovits Miklós 《Journal of Combinatorial Theory, Series B》1974,17(1):69-79
P. Turán has asked the following question:Let I12 be the graph determined by the vertices and edges of an icosahedron. What is the maximum number of edges of a graph Gn of n vertices if Gn does not contain I12 as a subgraph?We shall answer this question by proving that if n is sufficiently large, then there exists only one graph having maximum number of edges among the graphs of n vertices and not containing I12. This graph Hn can be defined in the following way:Let us divide n ? 2 vertices into 3 classes each of which contains [] or vertices. Join two vertices iff they are in different classes. Join two vertices outside of these classes to each other and to every vertex of these three classes. 相似文献
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Bondy conjectured [1] that: if G is a k-connected graph, where k ≥ 2, such that the degree-sum of any k + 1 independent vertices is at least m, then G contains a cycle of length at least: (n denotes the order of G). We prove here that this result is true. 相似文献
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G = 〈V(G), E(G)〉 denotes a directed graph without loops and multiple arrows. (G) denotes the set of all Hamiltonian circuits of G. Put H(n, r) = max{|E(G)|, |V(G)| = n, 1 ≤ |(G)| ≤ r}. Theorem: . Further, H(n, 2),…, H(n, 5) are given. 相似文献
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The graph G(P) of a polyhedron P has a node corresponding to each vertex of P and two nodes are adjacent in G(P) if and only if the corresponding vertices of P are adjacent on P. We show that if P ? n is a polyhedron, all of whose vertices have (0–1)-valued coordinates, then (i) if G(P) is bipartite, the G(P) is a hypercube; (ii) if G(P) is nonbipartite, then G(P) is hamilton connected. It is shown that if P ? n has (0–1)-valued vertices and is of dimension d (≤n) then there exists a polyhedron P′ ? d having (0–1)-valued vertices such that . Some combinatorial consequences of these results are also discussed. 相似文献
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Anne Penfold Street Earl Glen Whitehead 《Journal of Combinatorial Theory, Series A》1974,17(2):219-226
A subset S of a group G is said to be a sum-free set if . Such a set is maximal if for every sum-free set T ? G, we have |T| ? |S|. Here, we generalize this concept, defining a sum-free set S to be locally maximal if for every sum free set T such that S ? T ? G, we have S = T. Properties of locally maximal sum-free sets are studied and the sets are determined (up to isomorphism) for groups of small order. 相似文献
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Michael O Albertson 《Journal of Combinatorial Theory, Series B》1976,20(1):84-93
A subset of the vertices of a graph is independent if no two vertices in the subset are adjacent. The independence number α(G) is the maximum number of vertices in an independent set. We prove that if G is a planar graph on N vertices then . 相似文献
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Ronald Evans 《Journal of Number Theory》1973,5(2):108-115
Hecke proved analytically that when λ ≥ 2 or when , q ∈ Z, q ≥ 3, then is a fundamental region for the group G(λ) = 〈Sλ, T〉, where Sλ: τ → τ + λ and . He also showed that B(λ) fails to be a fundamental region for all other λ > 0 by proving that G(λ) is not discontinuous. We give an elementary proof of these facts and prove a related result concerning the distribution of G(λ)-equivalent points. 相似文献
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Denote by M(n) the smallest positive integer such that for every n-element monoid M there is a graph G with at most M(n) vertices such that End(G) is isomorphic to M. It is proved that . Moreover, for almost all n-element monoids M there is a graph G with at most 12 · n · log2n + n vertices such that End(G) is isomorphic to M. 相似文献
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This article is an attempt to study the following problem: Given a connected graph G, what is the maximum number of vertices of degree 1 of a spanning tree of G? For cubic graphs with n vertices, we prove that this number is bounded by and . 相似文献
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Jerrold R Griggs 《Journal of Combinatorial Theory, Series B》1983,34(1):22-39
Wei discovered that the independence number of a graph G is at least Σv(1 + d(v))?1. It is proved here that if G is a connected triangle-free graph on n ≥ 3 vertices and if G is neither an odd cycle nor an odd path, then the bound above can be increased by , where Δ is the maximum degree. This new bound is sharp for even cycles and for three other graphs. These results relate nicely to some algorithms for finding large independent sets. They also have a natural matrix theory interpretation. A survey of other known lower bounds on the independence number is presented. 相似文献
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Linda Lesmak 《Discrete Mathematics》1976,14(2):165-169
A graph G of order p ? 3 is called n-hamiltonian, 0 ? n ? p ? 3, if the removal of any m vertices, 0 ? m ? n, results in a hamiltonian graph. A graph G of order p ? 3 is defined to be n-hamiltonian, ?p ? n ? 1, if there exists ?n or fewer pairwise disjoint paths in G which collectively span G. Various conditions in terms of n and the degrees of the vertices of a graph are shown to be sufficient for the graph to be n-hamiltonian for all possible values of n. It is also shown that if G is a graph of order p ? 3 and , then G is n-hamiltonian. 相似文献
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A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X?{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X?{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number β(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G.A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely , is proved. 相似文献
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Let denote the complement of a graph G, and x(G), β1(G), β4(G), α0(G), α1(G) denote respectively the chromatic number, line-independence number, point-independence number, point-covering number, line-covering number of G, Nordhaus and Gaddum showed that for any graph G of order . Recently Chartrand and Schuster have given the corresponding inequalities for the independence numbers of any graph G. However, combining their result with Gallai's well known formula β1(G) + α1(G) = ?, one is not deduce the analogous bounds for the line-covering numbers of , since Gallai's formula holds only if G contains no isolated vertex. The purpose of this note is to improve the results of Chartland and Schuster for line-independence numbers for graphs where both contain no isolated vertices and thereby allowing us to use Gallai's formula to get the corresponding bounds for the line-covering numbers of G. 相似文献
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E. Turkel 《Linear algebra and its applications》1977,16(2):109-129
Sufficient conditions for the stability of multidimensional finite difference schemes are difficult to obtain. It is shown that for special families of amplification matrices G (A, B) a sufficient condition for power boundedness can be obtained by replacing the matrices by appropriate scalars, and so the problem is reduced to a scalar one. As one application it is shown that the Lax-Wendroff scheme in two dimensions is stable if |Au| + |Bu| ? 1 for all real unit vectors u. The Lax- Wendroff scheme with stabilizer does not always permit such large time steps. It is conjectured that the analysis for all symmetric hyperbolic schemes can be reduced to the scalar case. 相似文献
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Linda Lesniak 《Discrete Mathematics》1974,8(4):351-354
It is shown that if G is a graph of order p ≥ 2 such that deg u + deg v ≥ p ? 1 for all pairs u, v of nonadjacent vertices, then the edge-connectivity of G equals the minimum degree of G. Furthermore, if deg u + deg v ≥ p for all pairs u, v of nonadjacent vertices, then either p is even and G is isomorphic to or every minimum cutset of edges of G consists of the collection of edges incident with a vertex of least degree. 相似文献