共查询到20条相似文献,搜索用时 718 毫秒
1.
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
2.
Richard C OBrien 《Journal of Combinatorial Theory, Series B》1977,22(2):168-174
A proof is presented of the conjecture of Alspach and Pullman that for any digraph G on n ≥ 4 vertices, the path number of G is at most []. 相似文献
3.
We investigate those graphs Gn with the property that any tree on n vertices occurs as subgraph of Gn. In particular, we consider the problem of estimating the minimum number of edges such a graph can have. We show that this number is bounded below and above by and , respectively. 相似文献
4.
J.I. Hall 《Discrete Mathematics》1977,17(1):85-94
A strongly connected digraphGwithnvertices satisfying the condition that the sum of degrees for any two nonadjacent vertices is at least 2nis pancyclic unlessGis a tournament ornis even and 相似文献
5.
Colin McDiarmid 《Discrete Applied Mathematics》1983,5(1):123-132
When we wish to compute lower bounds for the chromatic number χ(G) of a graph G, it is of interest to know something about the ‘chromatic forcing number’ fχ(G), which is defined to be the least number of vertices in a subgraph H of G such that χ(H) = χ(G). We show here that for random graphs Gn,p with n vertices, fχ(Gn,p) is almost surely at least (?ε)n, despite say the fact that the largest complete subgraph of Gn,p has only about log n vertices. 相似文献
6.
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
7.
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
8.
Paul Erdős Norbert Sauer Jonathan Schaer Joel Spencer 《Journal of Combinatorial Theory, Series B》1975,18(2):180-183
The graph G has star number n if any n vertices of G belong to a subgraph which is a star. Let f(n, k) be the smallest number m such that the complete graph on m vertices can be factorized into k factors with star number n. In the present paper we prove that . 相似文献
9.
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
10.
D.R Woodall 《Journal of Combinatorial Theory, Series B》1978,25(2):184-186
A theorem is proved that is (in a sense) the best possible improvement on the following theme: If G is an undirected graph on n vertices in which for every non-empty subset S of the vertices of G, then G is Hamiltonian. 相似文献
11.
For finite graphs F and G, let NF(G) denote the number of occurrences of F in G, i.e., the number of subgraphs of G which are isomorphic to F. If and are families of graphs, it is natural to ask then whether or not the quantities NF(G), F∈, are linearly independent when G is restricted to . For example, if = {K1, K2} (where Kn denotes the complete graph on n vertices) and is the family of all (finite) trees, then of course NK1(T) ? NK2(T) = 1 for all T∈. Slightly less trivially, if = {Sn: n = 1, 2, 3,…} (where Sn denotes the star on n edges) and again is the family of all trees, then Σn=1∞(?1)n+1NSn(T)=1 for all T∈. It is proved that such a linear dependence can never occur if is finite, no F∈ has an isolated point, and contains all trees. This result has important applications in recent work of L. Lovász and one of the authors (Graham and Lovász, to appear). 相似文献
12.
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
13.
Jerrold R. Griggs 《Discrete Mathematics》1979,28(1):37-47
It is shown that the interval number of a graph on n vertices is at most , and this bound is best possible. This means that we can represent any graph on n vertices as an intersection graph in which the sets assigned to the vertices each consist of the union of at most finite closed intervals on the real line. This upper bound is attained by the complete bipartite graph K[n/2],[n/2]. 相似文献
14.
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
15.
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
16.
A signed graph based on F is an ordinary graph F with each edge marked as positive or negative. Such a graph is called balanced if each of its cycles includes an even number of negative edges. Psychologists are sometimes interested in the smallest number d=d(G) such that a signed graph G may be converted into a balanced graph by changing the signs of d edges. We investigate the number D(F) defined as the largest d(G) such that G is a signed graph based on F. We prove that for every graph F with n vertices and m edges. If F is the complete bipartite graph with t vertices in each part, then for some positive constant c. 相似文献
17.
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
18.
G is any simple graph with m edges and n vertices where these vertices have vertex degrees d(1)≥d(2)≥?≥d(n), respectively. General algorithms for the exact calculation of χ(G), the chromatic number of G, are almost always of ‘branch and bound’ type; this type of algorithm requires an easily constructed lower bound for χ(G). In this paper it is shown that, for a number of such bounds (many of which are described here for the first time), the bound does not exceed cl(G) where cl(G) is the clique number of G.In 1972 Myers and Liu showed that cl(G)≥n?(n?2m?n). Here we show that cl(G)≥ n?[n?(2m?n)(1+c2v)], where cv is the vertex degree coefficient of variation in G, that cl(G)≥ Bondy's constructive lower bound for χ(G), and that cl(G)≥n?(n?W(G)), where W(G) is the largest positive integer such that W(G)≤d(W(G)+1) [W(G)+1 is the Welsh and Powell upper bound for χ(G)]. It is also shown that cl(G)+>n?(n?L(G))≥n?(n?λ1) and that χ(G)≥ n?(n?λ1); λ1 is the largest eigenvalue of A, the adjacency matrix of G, and L(G) is a newly defined single-valued function of G similar to W(G) in its construction from the vertex degree sequence of G [L(G)≥W(G)].Finally, a ‘greedy’ lower bound for cl(G) is constructed from A and it is shown that this lower bound is never less than Bondy's bound; thereafter, for 150 random graphs with varying numbers of vertices and edge densities, the values of lower bounds given in this paper are listed, thereby illustrating that this last greedily obtained lower bound is almost always better than each of the other bounds. 相似文献
19.
In this paper we solve a conjecture of P. Erdös by showing that if a graph Gn has n vertices and at least edges, then G contains a cycle C2l of length 2l for every integer . Apart from the value of the constant this result is best possible. It is obtained from a more general theorem which also yields corresponding results in the case where Gn has only cn(log n)α edges (α ≥ 1). 相似文献
20.
R.Glenn Powers 《Discrete Mathematics》1982,41(3):271-279
Let G be a graph with chromatic number χ(G) and let t(G) be the minimum number of vertices in any color class among all χ(G)-vertex colorings of G. Let H′ be a connected graph and let H be a graph obtained by subdividing (adding extra vertices to) a fixed edge of H′. It is proved that if the order of H is sufficiently large, the ith Ramsey number ri(G, H) equals . 相似文献