共查询到20条相似文献,搜索用时 46 毫秒
1.
《Journal of Combinatorial Theory, Series B》1986,41(1):115-138
Let G be a graph drawn on a disc, and let the vertices of G on the boundary of the disc be s1, …, sk, t1, …, tk in some order. When are there k vertex-disjoint paths of G joining si and ti (1 ≤ i ≤ k), respectively? We give a structural characterization and a polynomial algorithm for this problem. We also solve the same question when the disc is replaced by a cylinder. 相似文献
2.
Haruko Okamura 《Graphs and Combinatorics》1987,3(1):159-189
Suppose thatk ≥ 1 is an odd integer, (s 1,t 1),..., (s k> ,t k ) are pairs of vertices of a graphG andλ(s i ,t i ) is the maximal number of edge-disjoint paths betweens i andt i . We prove that ifλ(s i ,t i )≥ k (1≤ i ≤ k) and |{s 1,...s k ,t 1,...,t k }| ≤ 6, then there exist edge-disjoint pathsP 1,...,P k such thatP i has endss i andt i (1≤ i ≤ k). 相似文献
3.
Tomio Hirata Kiyohito Kubota Osami Saito 《Journal of Combinatorial Theory, Series B》1984,36(1):85-94
For a pair (s, t) of vertices of a graph G, let λG(s, t) denote the maximal number of edge-disjoint paths between s and t. Let (s1, t1), (s2, t2), (s3, t3) be pairs of vertices of G and k > 2. It is shown that if λG(si, ti) ≥ 2k + 1 for each i = 1, 2, 3, then there exist 2k + 1 edge-disjoint paths such that one joins s1 and t1, another joins s2 and t2 and the others join s3 and t3. As a corollary, every (2k + 1)-edge-connected graph is weakly (k + 2)-linked for k ≥ 2, where a graph is weakly k-linked if for any k vertex pairs (si, ti), 1 ≤ i ≤ k, there exist k edge-disjoint paths P1, P2,…, Pk such that Pi joins si and ti for i = 1, 2,…, k. 相似文献
4.
Hong Wang 《Graphs and Combinatorics》1994,10(2-4):271-281
Letk and s be two positive integers with s≥3. LetG be a graph of ordern ≥sk. Writen =qk + r, 0 ≤r ≤k - 1. Suppose thatG has minimum degree at least (s - l)k. Then G containsk independent cyclesC 1,C 2,...,C k such thats ≤l(C i ) ≤q for 1 ≤i ≤r arnds ≤l(C i ) ≤q + 1 fork -r <i ≤k, where l(Ci) denotes the length ofC i . 相似文献
5.
Let G1, G2,. …, Gt be an arbitrary t-edge coloring of Kn, where for each i ∈ {1,2, …, t}, Gi is the spanning subgraph of Kn consisting of all edges colored with the ith color. The irredundant Ramsey number s(q1, q2, …, qt) is defined as the smallest integer n such that for any t-edge coloring of Kn, i has an irredundant set of size qi for at least one i ∈ {1,2, …,t}. It is proved that s(3,3,3) = 13, a result that improves the known bounds 12 ≤ s(3,3,3) ≤ 14. 相似文献
6.
Jiuying Dong 《Journal of Applied Mathematics and Computing》2010,34(1-2):485-493
The theory of vertex-disjoint cycles and 2-factor of graphs has important applications in computer science and network communication. For a graph G, let σ 2(G):=min?{d(u)+d(v)|uv ? E(G),u≠v}. In the paper, the main results of this paper are as follows:
- Let k≥2 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k of length at most four such that v i ∈V(C i ) for all 1≤i≤k.
- Let k≥1 be an integer and G be a graph of order n≥3k, if σ 2(G)≥n+2k?2, then for any set of k distinct vertices v 1,…,v k , G has k vertex-disjoint cycles C 1,C 2,…,C k such that:
- v i ∈V(C i ) for all 1≤i≤k.
- V(C 1)∪???∪V(C k )=V(G), and
- |C i |≤4, 1≤i≤k?1.
7.
A. V. Karzanov 《Mathematical Programming》1985,32(2):183-198
Suppose thatG is an undirected graph whose edges have nonnegative integer-valued lengthsl(e), and that {s 1,t 1},?, {s m ,t m } are pairs of its vertices. Can one assign nonnegative weights to the cuts ofG such that, for each edgee, the total weight of cuts containinge does not exceedl(e) and, for eachi, the total weight of cuts ‘separating’s i andt i is equal to the distance (with respect tol) betweens i andt i ? Using linear programming duality, it follows from Papernov's multicommodity flow theorem that the answer is affirmative if the graph induced by the pairs {s 1,t 1},?, {s m ,t m } is one of the following: (i) the complete graph with four vertices, (ii) the circuit with five vertices, (iii) a union of two stars. We prove that if, in addition, each circuit inG has an even length (with respect tol) then there exists a suitable weighting of the cuts with the weights integer-valued; moreover, an algorithm of complexity O(n 3) (n is the number of vertices ofG) is developed for solving such a problem. Also a class of metrics decomposable into a nonnegative linear combination of cut-metrics is described, and it is shown that the separation problem for cut cones isNP-hard. 相似文献
8.
Yolandé Jacobs Elizabeth Jonck Ernst J. Joubert 《Central European Journal of Mathematics》2013,11(7):1344-1357
Let G = (V, E) be a simple graph of order n and i be an integer with i ≥ 1. The set X i ? V(G) is called an i-packing if each two distinct vertices in X i are more than i apart. A packing colouring of G is a partition X = {X 1, X 2, …, X k } of V(G) such that each colour class X i is an i-packing. The minimum order k of a packing colouring is called the packing chromatic number of G, denoted by χρ(G). In this paper we show, using a theoretical proof, that if q = 4t, for some integer t ≥ 3, then 9 ≤ χρ(C 4 □ C q ). We will also show that if t is a multiple of four, then χρ(C 4 □ C q ) = 9. 相似文献
9.
《European Journal of Combinatorics》2005,26(3-4):309-324
A graph is said to be k-linked if it has at least 2k vertices and for every sequence s1,…,sk,t1,…,tk of distinct vertices there exist disjoint paths P1,…,Pk such that the ends of Pi are si and ti. Bollobás and Thomason showed that if a simple graph G on n vertices is 2k-connected and G has at least 11kn edges, then G is k-linked. We give a relatively simple inductive proof of the stronger statement that 8kn edges and 2k-connectivity suffice, and then with more effort improve the edge bound to 5kn. 相似文献
10.
For positive integers n1, n2, …, nI and graphs GI+1, GI+2, …, Gk, 1 ≤ / < k, the mixed Ramsey number χ(n1, …, n1, GI+1, …, Gk) is define as the least positive integer p such that for each factorization Kp = F1⊕ … ⊕ F⊕ FI+1⊕ … ⊕ Fk, it it follows that χ(Fi) ≥ ni for some i, 1 ? i ? l, or Gi is a subgraph of Fi for some i, l < i ? k. Formulas are presented for maxed Ramsey numbers in which the graphs GI+1, GI+2, …, Gk are connected, and in which k = I+1 and GI+1 is arbitray. 相似文献
11.
《Discrete Applied Mathematics》1987,18(2):227-233
We consider the problem of job shop scheduling with m machines and n jobs Ji, each consisting of li unit time operations. There are s distinct resources Rh and a quantity qh available of each one. The execution of the j-th operation of Ji requires the presence of uijh units of Rh, 1 ≤i≤n, 1 ≤j≤li, and 1 ≤h≤s. In addition, each Ji has a release date ri, that is Ji cannot start before time ri. We describe algorithms for finding schedules having minimum length or sum of completion times of the jobs. Let l=max{li} and u=|{uijh}|. If m, u and l are fixed, then both algorithms terminate within polynomial time. 相似文献
12.
Let G be a group, t an element distinct from G, and r(t) = g 1 t l 1 …g k t l k ∈ G* ?t?, where each g i is an element of G order greater than 2, and the l i are nonzero integers such that l 1 + l 2+…+l k ≠ 0 and |l i | ≠ |l j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ?G*t | r(t)? is injective. This together with previous results show that the natural map from G is injective for k ≥ 1. 相似文献
13.
F. M. Dong K. M. Koh K. L. Teo C. H. C. Little M. D. Hendy 《Journal of Graph Theory》2001,37(1):48-77
Given a graph G and an integer k ≥ 1, let α(G, k) denote the number of k‐independent partitions of G. Let ???s(p,q) (resp., ??2?s(p,q)) denote the family of connected (resp., 2‐connected) graphs which are obtained from the complete bipartite graph Kp,q by deleting a set of s edges, where p ≥ q ≥ 2. This paper first gives a sharp upper bound for α(G,3), where G ∈ ?? ?s(p,q) and 0 ≤ s ≤ (p ? 1)(q ? 1) (resp., G ∈ ?? 2?s(p,q) and 0 ≤ s ≤ p + q ? 4). These bounds are then used to show that if G ∈ ?? ?s(p,q) (resp., G ∈ ?? 2?s (p,q)), then the chromatic equivalence class of G is a subset of the union of the sets ???si(p+i,q?i) where max and si = s ? i(p?q+i) (resp., a subset of ??2?s(p,q), where either 0 ≤ s ≤ q ? 1, or s ≤ 2q ? 3 and p ≥ q + 4). By applying these results, we show finally that any 2‐connected graph obtained from Kp,q by deleting a set of edges that forms a matching of size at most q ? 1 or that induces a star is chromatically unique. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 48–77, 2001 相似文献
14.
Given two nonnegative integers s and t, a graph G is (s,t)-supereulerian if for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y|≤t, there is a spanning eulerian subgraph H of G that contains X and avoids Y. We prove that if G is connected and locally k-edge-connected, then G is (s,t)-supereulerian, for any pair of nonnegative integers s and t with s+t≤k−1. We further show that if s+t≤k and G is a connected, locally k-edge-connected graph, then for any disjoint sets X,Y⊂E(G) with |X|≤s and |Y≤t, there is a spanning eulerian subgraph H that contains X and avoids Y, if and only if G−Y is not contractible to K2 or to K2,l with l odd. 相似文献
15.
A finite group G is said to be a B(n, k) group if for any n-element subset {a 1,…, a n } of G, |{a i a j |1 ≤ i, j ≤ n}| ≤k. In this article, we give characterizations of the B(5, 19) 2-groups, and the B(6, k) 2-groups for 21 ≤ k ≤ 28. 相似文献
16.
Let k≥1 be an integer and G=(V 1,V 2;E) a bipartite graph with |V 1|=|V 2|=n such that n≥2k+2. Our result is as follows: If $d(x)+d(y)\geq \lceil\frac{4n+k}{3}\rceil$ for any nonadjacent vertices x∈V 1 and y∈V 2, then for any k distinct vertices z 1,…,z k , G contains a 2-factor with k+1 cycles C 1,…,C k+1 such that z i ∈V(C i ) and l(C i )=4 for each i∈{1,…,k}. 相似文献
17.
M. R. Darafsheh N. S. Karamzadeh A. R. Moghaddamfar 《Journal of Applied Mathematics and Computing》2006,21(1-2):437-450
IfG is a finite group, we define its prime graph Г(G), as follows: its vertices are the primes dividing the order ofG and two verticesp, q are joined by an edge, if there is an element inG of orderpq. We denote the set of all the connected components of the graph Г(G) by T(G)= {πi(G), fori = 1,2, …,t(G)}, where t(G) is the number of connected components of Г(G). We also denote by π(n) the set of all primes dividingn, wheren is a natural number. Then ¦G¦ can be expressed as a product of m1, m2, …, mt(G), where mi’s are positive integers with π(mi) = πi. Thesem i ’s are called the order components ofG. LetOC(G) := {m 1,m 2, …,m t (G)} be the set of order components ofG. In this paper we prove that, if G is a finite group andOC(G) =OC(M), where M is a finite simple group witht(M) ≥ 2, thenG is neither Frobenius nor 2-Frobenius. 相似文献
18.
Andreas Huck 《Journal of Graph Theory》1992,16(6):571-589
We consider finite undirected loopless graphs G in which multiple edges are possible. For integers k,l ≥ 0 let g(k, l) be the minimal n ≥ 0 with the following property: If G is an n-edge-connected graph, s1, ?,sk, t1, ?,tk are vertices of G, and f1, ?,fl, g1, ?,gl, are pairwise distinct edges of G, then for each i = 1, ?, k there exists a path Pi in G, connecting si and ti and for each i = 1, ?,l there exists a cycle Ci in G containing fi and gi such that P1, ?,Pk, C1, ?, Cl are pairwise edge-disjoint. We give upper and lower bounds for g(k, l). 相似文献
19.
In this paper we consider the general Ramsey number problem for paths when the complete graph is colored with k colors. Specifically, given paths Pi1, Pi2,…, Pik with i1, i2,…, ik vertices, we determine for certain ij (1 ≤ j ≤ k) the smallest positive integer n such that a k coloring of the complete graph Kn contains, for some l, a Pil in the lth color. For k = 3, given i2, i3, the problem is solved for all but a finite number of values of i1. The procedure used in the proof uses an improvement of an extremal theorem for paths by P. Erdös and T. Gallai. 相似文献
20.
Disjoint triangles and quadrilaterals in a graph 总被引:1,自引:0,他引:1
Jin Yan 《Discrete Mathematics》2008,308(17):3930-3937
Let G be a simple graph of order n and s and k be two positive integers. Brandt et al. obtained the following result: If s?k, n?3s+4(k-s) and σ2(G)?n+s, then G contains k disjoint cycles C1,…,Ck satisfying |Ci|=3 for 1?i?s and |Ci|?4 for s<i?k. In the above result, the length of Ci is not specified for s<i?k. We get a result specifying the length of Ci for each s<i?k if n?3s+4(k-s)+3. 相似文献