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1.
(3,k)-Factor-Critical Graphs and Toughness 总被引:1,自引:0,他引:1
A graph is (r,k)-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. an r-regular spanning subgraph). Let t(G) denote the toughness of graph G. In this paper, we show that if t(G)≥4, then G is (3,k)-factor-critical for every non-negative integer k such that n+k even, k<2 t(G)−2 and k≤n−7.
Revised: September 21, 1998 相似文献
2.
P. Katerinis 《Journal of Graph Theory》1987,11(1):53-58
Upper bounds are placed on the order of a k-regular m-connected graph G that produce a lower bound on the number of independent edges in G. As a corollary, we obtain the order of a smallest k-regular m-connected graph which has no 1-factor. 相似文献
3.
Hong Wang 《Graphs and Combinatorics》2001,17(1):177-183
Let G=(V
1,V
2;E) be a bipartite graph with 2k≤m=|V
1|≤|V
2|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs.
Received: December 9, 1998 Final version received: June 2, 1999 相似文献
4.
(2,k)-Factor-Critical Graphs and Toughness 总被引:1,自引:0,他引:1
A graph is (r,k)-factor-critical if the removal of any set of k vertices results in a graph with an r-factor (i.e. with an r-regular spanning subgraph). We show that every τ-tough graph of order n with τ≥2 is (2,k)-factor-critical for every non-negative integer k≤min{2τ−2, n−3}, thus proving a conjecture as well as generalizing the main result of Liu and Yu in [4].
Received: December 16, 1996 / Revised: September 17, 1997 相似文献
5.
An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N
G−e
(x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present
paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4). 相似文献
6.
A k-tree of a graph is a spanning tree with maximum degree at most k. We give sufficient conditions for a graph G to have a k-tree with specified leaves: Let k,s, and n be integers such that k≥2, 0≤s≤k, and n≥s+1. Suppose that (1) G is (s+1)-connected and the degree sum of any k independent vertices of G is at least |G|+(k−1)s−1, or (2) G is n-connected and the independence number of G is at most (n−s)(k−1)+1. Then for any s specified vertices of G, G has a k-tree containing them as leaves. We also discuss the sharpness of the results.
This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Encouragement
of Young Scientists, 15740077, 2005
This research was partially supported by the Japan Society for the Promotion of Science for Young Scientists. 相似文献
7.
Zhi-quanHu FengTian 《应用数学学报(英文版)》2003,19(1):97-106
A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G)≥3κ-1/2 or δ(G)≥5κ-4.Several known results are generalized. 相似文献
8.
Jian-pingOu Fu-jiZhang 《应用数学学报(英文版)》2003,19(3):505-510
An m-restricted edge cut is an edge cut that separates a connected graph into a disconnected one with no components having order less than m. m-restrict edge connectivity λm is the cardinality of a minimum m-restricted edge cut. Let G be a connected k-regular graph of order at least 2m that contains m-restricted edge cuts and X be a subgraph of G. Let θ(X) denote the number of edges with one end in X and the other not in X and ξm=min{θ(X) ;X is a connected vertex-induced subgraph of order m}.It is proved in this paper that if G has girth at least m/2 2,then λm≤ξm.The upper bound of λm is sharp. 相似文献
9.
Some results on R
2-edge-connectivity of even regular graphs 总被引:1,自引:0,他引:1
XuJunming 《高校应用数学学报(英文版)》1999,14(3):366-370
Let G be a connected k(≥3)-regular graph with girth g. A set S of the edges in G is called an Rredge-cut if G-S is disconnected and comains neither an isolated vertex nor a one-degree vertex. The R2-edge-connectivity of G, denoted by λ^n(G), is the minimum cardinality over all R2-edge-cuts, which is an important measure for fault-tolerance of computer interconnection networks. In this paper, λ^n(G)=g(2k-2) for any 2k-regular connected graph G (≠K5) that is either edge-transitive or vertex-transitive and g≥5 is given. 相似文献
10.
The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d
G
(x) + d
G
(y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d
G
(u) ≥ k. 相似文献
11.
For an integer k > 0, a graph G is k-triangular if every edge of G lies in at least k distinct 3-cycles of G. In (J Graph Theory 11:399–407 (1987)), Broersma and Veldman proposed an open problem: for a given positive integer k, determine the value s for which the statement “Let G be a k-triangular graph. Then L(G), the line graph of G, is s-hamiltonian if and only L(G) is (s + 2)-connected” is valid. Broersma and Veldman proved in 1987 that the statement above holds for 0 ≤ s ≤ k and asked, specifically, if the statement holds when s = 2k. In this paper, we prove that the statement above holds for 0 ≤ s ≤ max{2k, 6k − 16}. 相似文献
12.
A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph
Kazuhiro Suzuki 《Graphs and Combinatorics》2006,22(2):261-269
We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. 相似文献
13.
Camino Balbuena Martín Cera Pedro García-Vázquez Juan Carlos Valenzuela 《数学学报(英文版)》2011,27(11):2085-2100
For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K
(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. 相似文献
14.
HaoZHAO GuiZhenLIU XiaoXiaYAN 《数学学报(英文版)》2005,21(2):413-422
Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integervalued functions defined on V(G) such that 2k - 2 ≤g(x)≤f(x) for all x∈V(G). Let H be a subgraph of G with mk edges. In this paper, it is proved that every (mg m-1,mf-m 1)-graph G has (g, f)-factorizations randomly k-orthogonal to H under some special conditions. 相似文献
15.
An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least
2; a graph G is super restricted edge connected if G−S contains an isolated edge for every minimum restricted edge cut S of G. It is proved in this paper that k-regular connected graph G is super restricted edge connected if k > |V(G)|/2+1. The lower bound on k is exemplified to be sharp to some extent. With this observation, we determined the number of edge cuts of size at most 2k−2 of these graphs.
Supported by NNSF of China (10271105); Ministry of Science and Technology of Fujian (2003J036); Education Ministry of Fujian
(JA03147) 相似文献
16.
Anna Draganova 《Czechoslovak Mathematical Journal》2009,59(1):51-60
For any nontrivial connected graph F and any graph G, the F-degree of a vertex v in G is the number of copies of F in G containing v. G is called F-continuous if and only if the F-degrees of any two adjacent vertices in G differ by at most 1; G is F-regular if the F-degrees of all vertices in G are the same. This paper classifies all P
4-continuous graphs with girth greater than 3. We show that for any nontrivial connected graph F other than the star K
1,k
, k ⩾ 1, there exists a regular graph that is not F-continuous. If F is 2-connected, then there exists a regular F-continuous graph that is not F-regular.
相似文献
17.
On (g, f)-Uniform Graphs 总被引:3,自引:0,他引:3
Gui-zhenLiu YanLiu 《应用数学学报(英文版)》2005,21(1):67-76
A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple sufficient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b. 相似文献
18.
The k-core of a graph is the largest subgraph of minimum degree at least k. We show that for k sufficiently large, the threshold for the appearance of a k-regular subgraph in the Erdős-Rényi random graph model G(n,p) is at most the threshold for the appearance of a nonempty (k+2)-core. In particular, this pins down the point of appearance of a k-regular subgraph to a window for p of width roughly 2/n for large n and moderately large k. The result is proved by using Tutte’s necessary and sufficient condition for a graph to have a k-factor. 相似文献
19.
Dr. Matthias Kriesell 《Combinatorica》2006,26(3):277-314
A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism).
Here we prove this conjecture. 相似文献
20.
In this paper we study three-color Ramsey numbers. Let K
i,j
denote a complete i by j bipartite graph. We shall show that (i) for any connected graphs G
1, G
2 and G
3, if r(G
1, G
2)≥s(G
3), then r(G
1, G
2, G
3)≥(r(G
1, G
2)−1)(χ(G
3)−1)+s(G
3), where s(G
3) is the chromatic surplus of G
3; (ii) (k+m−2)(n−1)+1≤r(K
1,k
, K
1,m
, K
n
)≤ (k+m−1)(n−1)+1, and if k or m is odd, the second inequality becomes an equality; (iii) for any fixed m≥k≥2, there is a constant c such that r(K
k,m
, K
k,m
, K
n
)≤c(n/logn), and r(C
2m
, C
2m
, K
n
)≤c(n/logn)
m/(m−1)
for sufficiently large n.
Received: July 25, 2000 Final version received: July 30, 2002
RID="*"
ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of
education ministry of China, and the foundations of Jiangsu Province
Acknowledgments. The authors are grateful to the referee for his valuable comments.
AMS 2000 MSC: 05C55 相似文献