共查询到20条相似文献,搜索用时 250 毫秒
1.
Let be a family of subsets of S and let G be a graph with vertex set such that: (xA, xB) is an edge iff . The family is called a set representation of the graph G.It is proved that the problem of finding minimum k such that G can be represented by a family of sets of cardinality at most k is NP-complete. Moreover, it is NP-complete to decide whether a graph can be represented by a family of distinct 3-element sets.The set representations of random graphs are also considered. 相似文献
2.
Cai Mao-cheng 《Discrete Mathematics》1982,41(3):229-234
Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) ; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph Kk,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove and characterize all minimally k-connected graphs of order n and size . 相似文献
3.
Daniel A Marcus 《Journal of Combinatorial Theory, Series B》1981,30(1):21-31
It is shown that every k-edge-connected digraph with m edges and n vertices contains a spanning connected subgraph having at most edges. When k = 2 the bound is improved to , which implies that a 2-edge-connected digraph is connected by less than 70% of its edges. Examples are given which require almost two-thirds of the edges to connect all vertices. 相似文献
4.
Jacqueline Ayel 《Journal of Combinatorial Theory, Series B》1982,32(2):223-228
Let (itk, p) denote the class of k-partite graphs, where each part is a stable set of cardinality p and where the edges between any pair of stable sets are those of a perfect matching. Maru?i? has conjectured that if G belongs to (k, p) and is connected then G is hamiltonian. It is proved that the conjecture is true for k ≤ 3 or p ≤ 3; but for k ≥ 4 and p ≥ 4 a non-hamiltonian connected graph in (k, p) is constructed. 相似文献
5.
Following a conjecture of P. Erdös, we show that if is a family of k-subsets of and n-set no two of which intersect in exactly l elements then for k ? 2l + 2 and n sufficiently large || ? (k ? l ? 1n ? l ? 1) with equality holding if and only if consists of all the k-sets containing a fixed (l + 1)-set. In general we show || ? dknmax;{;l,k ? l ? 1};, where dk is a constant depending only on k. These results are special cases of more general theorems (Theorem 2.1–2.3). 相似文献
6.
A function diagram (f-diagram) D consists of the family of curves {ñ} obtained from n continuous functions . We call the intersection graph of D a function graph (f-graph). It is shown that a graph G is an f-graph if and only if its complement ? is a comparability graph. An f-diagram generalizes the notion of a permulation diagram where the fi are linear functions. It is also shown that G is the intersection graph of the concatenation of ?k permutation diagrams if and only if the partial order dimension of . Computational complexity results are obtained for recognizing such graphs. 相似文献
7.
Paul Terwilliger 《Journal of Combinatorial Theory, Series B》1982,32(2):182-188
It is shown that any bipartite distance-regular graph with finite valency k and at least one cycle is finite, with diameter d and girth g satisfying . In particular, the number of bipartite distance-regular graphs with fixed valency and girth is finite. 相似文献
8.
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
9.
Rüdiger Schmidt 《Discrete Mathematics》1979,27(1):93-97
A matroidal family is a set ≠ ? of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way. 相似文献
10.
László Miklós Lovász Carsten Thomassen Yezhou Wu Cun-Quan Zhang 《Journal of Combinatorial Theory, Series B》2013
The main theorem of this paper provides partial results on some major open problems in graph theory, such as Tutte?s 3-flow conjecture (from the 1970s) that every 4-edge connected graph admits a nowhere-zero 3-flow, the conjecture of Jaeger, Linial, Payan and Tarsi (1992) that every 5-edge-connected graph is Z3-connected, Jaeger?s circular flow conjecture (1984) that for every odd natural number k?3, every (2k−2)-edge-connected graph has a modulo k -orientation, etc. It was proved recently by Thomassen that, for every odd number k?3, every (2k2+k)-edge-connected graph G has a modulo k-orientation; and every 8-edge-connected graph G is Z3-connected and admits therefore a nowhere-zero 3-flow. In the present paper, Thomassen?s method is refined to prove the following: For every odd number k?3, every (3k−3)-edge-connected graph has a modulo k-orientation. As a special case of the main result, every 6-edge-connected graph is Z3-connected and admits therefore a nowhere-zero 3-flow. Note that it was proved by Kochol (2001) that it suffices to prove the 3-flow conjecture for 5-edge-connected graphs. 相似文献
11.
David W Matula 《Journal of Combinatorial Theory, Series B》1978,24(1):1-13
A k-block is a maximal k-vertex-connected subgraph, and a k-block which does not contain a (k + 1)-block is an ultrablock. It is shown that the maximum total number of k-blocks for all k ≥ 1 in any p-vertex graph is [], and the maximum number of ultrablocks in any p-vertex graph having maximum subgraph connectivity κ? is . In contrast to the linear growth rate of the maximum number of k-blocks in a p-vertex graph, it is shown that the maximum number of critical k-vertex-connected subgraphs of an ultrablock of connectivity k can grow exponentially with p. 相似文献
12.
Berge's strong perfect-graph conjecture states that a graph is perfect iff it has neither C2n+1 nor , n ≥ 2 as an induced subgraph. In this note we establish the validity of this conjecture for (K4?e)-free graphs. 相似文献
13.
Paul Terwilliger 《Discrete Mathematics》1982,41(3):295-302
We find lower bounds on eigenvalue multiplicities for highly symmetric graphs. In particular we prove:Theorem 1. If Γ is distance-regular with valency k and girth g (g?4), and λ (λ≠±?k) is an eigenvalue of Γ, then the multiplicity of λ is at least if g≡0 or 1 (mod 4), if g≡2 or 3 (mod 4) where [ ] denotes integer part. Theorem 2. If the automorphism group of a regular graph Γ with girth g (g?4) and valency k acts transitively on s-arcs for some s, , then the multiplicity of any eigenvalue λ (λ≠±?k) is at least if s is even, if s is odd. 相似文献
14.
Peter J. Slater 《Discrete Mathematics》1981,34(2):185-193
The concept of a k-sequential graph is presented as follows. A graph G with ∣V(G)∪ E(G)∣=t is called k-sequential if there is a bijection such that for each edgein E(G) one has. A graph that is 1-sequential is called simply sequential, and, in particular the author has conjectured that all trees are simply sequential. In this paper an introductory study of k-sequential graphs is made. Further, several variations on the problems of gracefully or sequentially numbering the elements of a graph are discussed. 相似文献
15.
The path-connectivity of a graph G is the maximal k for which between any k pairs of vertices there are k edge-disjoint paths (one between each pair). An upper bound for the path-connectivity of nq(q<1) separable graphs [6] is shown to exist.If the edge-connectivity of a graph is KE then between any two pairs of vertices and for every there exists a t?t′?t+1 such that there are t′ paths between the first pair and between the second pair. All paths are edge-disjoint. 相似文献
16.
Distance-regular graphs of diameter three are of three (almost distinct) kinds: primitive, bipartite, and antipodal. An antipodal graph of diameter three is just an r-fold covering of a complete graph Kk+1 for some r?k. Its intersection array and spectrum are determined by the parameters r, k together with the number c of 2-arcs joining any two vertices at distance two. Most such graphs have girth three. In this note we consider antipodal distance-regular graphs of diameter three and girth ? 4. If r=2, then the only graphs are “Kk+1, k+1 minus a 1-factor.” We therefore assume r?3. The graphs with c=1 necessarily have r=k and were classified in lsqb3rsqb. We prove the inequality (Theorem 2), list the feasible parameter sets when c=2 or 3 (Corollary 1), and conclude that every 3-fold or 4-fold antipodal covering of a complete graph has girth three (Corollary 2). 相似文献
17.
F.R.K. Chung 《Journal of Combinatorial Theory, Series A》1983,35(3):252-262
Suppose is a collection of 3-subsets of {1,2,…,n}. The problem of determining the least integer with the property that if then contains a k-star (i.e., k 3-sets such that the intersection of any pair of them consists of exactly the same element) is studied. It is proved that, for k odd, and, for k even, . 相似文献
18.
Yahya Ould Hamidoune 《Journal of Combinatorial Theory, Series B》1981,30(1):108-112
A conjecture of Slater states that Kh + 1 is the unique k-critically h-connected noncomplete graph for 2k > h. We prove here that there is no k-critically h-connected connected graph with order . We prove also that there is no k-critically h-connected line graph for 2k > h. The last result was conjectured by Maurer and Slater. We apply in our proofs a method introduced by Mader. 相似文献
19.
20.
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