共查询到20条相似文献,搜索用时 31 毫秒
1.
Tomoki Yamashita 《Discrete Mathematics》2008,308(24):6584-6587
Let G be a graph. For S⊂V(G), let Δk(S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ2(S)≥d for every independent set S of order κ(G)+1, then G has a cycle of length at least min{d,|V(G)|}. (2) Let G be a 2-connected graph and X a subset of V(G). If Δ2(S)≥|V(G)| for every independent set S of order κ(X)+1 in G[X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles. 相似文献
2.
Ioan Tomescu 《Discrete Mathematics》2009,309(9):2745-788
The degree distance of a connected graph, introduced by Dobrynin, Kochetova and Gutman, has been studied in mathematical chemistry. In this paper some properties of graphs having minimum degree distance in the class of connected graphs of order n and size m≥n−1 are deduced. It is shown that any such graph G has no induced subgraph isomorphic to P4, contains a vertex z of degree n−1 such that G−z has at most one connected component C such that |C|≥2 and C has properties similar to those of G.For any fixed k such that k=0,1 or k≥3, if m=n+k and n≥k+3 then the extremal graph is unique and it is isomorphic to K1+(K1,k+1∪(n−k−3)K1). 相似文献
3.
Shinya Fujita 《Discrete Mathematics》2009,309(11):3534-3540
Let k,n be integers with 2≤k≤n, and let G be a graph of order n. We prove that if max{dG(x),dG(y)}≥(n−k+1)/2 for any x,y∈V(G) with x≠y and xy∉E(G), then G has k vertex-disjoint subgraphs H1,…,Hk such that V(H1)∪?∪V(Hk)=V(G) and Hi is a cycle or K1 or K2 for each 1≤i≤k, unless k=2 and G=C5, or k=3 and G=K1∪C5. 相似文献
4.
In this paper, we give sufficient conditions for a graph to have degree bounded trees. Let G be a connected graph and . We denote by the minimum value of the degree sum in G of any k pairwise nonadjacent vertices of A, and by the number of components of the subgraph of G induced by . Our main results are the following: (i) If , then G contains a tree T with maximum degree ⩽k and . (ii) If , then G contains a spanning tree T with for any . These are generalizations of the result by S. Win [S. Win, Existenz von Gerüsten mit Vorgeschriebenem Maximalgrad in Graphen, Abh. Math. Seminar Univ. Humburg 43 (1975) 263–267] and degree conditions are sharp. 相似文献
5.
Fan [G. Fan, Distribution of cycle lengths in graphs, J. Combin. Theory Ser. B 84 (2002) 187-202] proved that if G is a graph with minimum degree δ(G)≥3k for any positive integer k, then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if δ(G)≥3k+1, then |E(Ck)|−|E(Ck−1)|=2. In this paper, we generalize Fan’s result, and show that if we let G be a graph with minimum degree δ(G)≥3, for any positive integer k (if k≥2, then δ(G)≥4), if dG(u)+dG(v)≥6k−1 for every pair of adjacent vertices u,v∈V(G), then G contains k+1 cycles C0,C1,…,Ck such that k+1<|E(C0)|<|E(C1)|<?<|E(Ck)|, |E(Ci)−E(Ci−1)|=2, 1≤i≤k−1, and 1≤|E(Ck)|−|E(Ck−1)|≤2, and furthermore, if dG(u)+dG(v)≥6k+1, then |E(Ck)|−|E(Ck−1)|=2. 相似文献
6.
Florian Pfender 《Discrete Mathematics》2009,309(9):2922-2924
We show that line graphs G=L(H) with σ2(G)≥7 contain cycles of all lengths k, 2rad(H)+1≤k≤c(G). This implies that every line graph of such a graph with 2rad(H)≥Δ(H) is subpancyclic, improving a recent result of Xiong and Li. The bound on σ2(G) is best possible. 相似文献
7.
Ioan Tomescu 《Discrete Applied Mathematics》2010,158(15):1714-1717
The concept of degree distance of a connected graph G is a variation of the well-known Wiener index, in which the degrees of vertices are also involved. It is defined by D′(G)=∑x∈V(G)d(x)∑y∈V(G)d(x,y), where d(x) and d(x,y) are the degree of x and the distance between x and y, respectively. In this paper it is proved that connected graphs of order n≥4 having the smallest degree distances are K1,n−1,BS(n−3,1) and K1,n−1+e (in this order), where BS(n−3,1) denotes the bistar consisting of vertex disjoint stars K1,n−3 and K1,1 with central vertices joined by an edge. 相似文献
8.
Kenta Ozeki 《Discrete Mathematics》2009,309(13):4266-4269
Win, in 1975, and Jackson and Wormald, in 1990, found the best sufficient conditions on the degree sum of a graph to guarantee the properties of “having a k-tree” and “having a k-walk”, respectively. The property of “being prism hamiltonian” is an intermediate property between “having a 2-tree” and “having a 2-walk”. Thus, it is natural to ask what is the best degree sum condition for graphs to be prism hamiltonian. As an answer to this problem, in this paper, we show that a connected graph G of order n with σ3(G)≥n is prism hamiltonian. The degree sum condition “σ3(G)≥n” is best possible. 相似文献
9.
Let G be an (m+2)-graph on n vertices, and F be a linear forest in G with |E(F)|=m and ω1(F)=s, where ω1(F) is the number of components of order one in F. We denote by σ3(G) the minimum value of the degree sum of three vertices which are pairwise non-adjacent. In this paper, we give several σ3 conditions for a dominating cycle or a hamiltonian cycle passing through a linear forest. We first prove that if σ3(G)≥n+2m+2+max{s−3,0}, then every longest cycle passing through F is dominating. Using this result, we prove that if σ3(G)≥n+κ(G)+2m−1 then G contains a hamiltonian cycle passing through F. As a corollary, we obtain a result that if G is a 3-connected graph and σ3(G)≥n+κ(G)+2, then G is hamiltonian-connected. 相似文献
10.
For a graph G, let σ2(G) denote the minimum degree sum of two nonadjacent vertices (when G is complete, we let σ2(G)=∞). In this paper, we show the following two results: (i) Let G be a graph of order n≥4k+3 with σ2(G)≥n and let F be a matching of size k in G such that G−F is 2-connected. Then G−F is hamiltonian or G≅K2+(K2∪Kn−4) or ; (ii) Let G be a graph of order n≥16k+1 with σ2(G)≥n and let F be a set of k edges of G such that G−F is hamiltonian. Then G−F is either pancyclic or bipartite. Examples show that first result is the best possible. 相似文献
11.
Let G be a graph of order n and maximum degree Δ(G) and let γt(G) denote the minimum cardinality of a total dominating set of a graph G. A graph G with no isolated vertex is the total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G−v is less than the total domination number of G. We call these graphs γt-critical. For any γt-critical graph G, it can be shown that n≤Δ(G)(γt(G)−1)+1. In this paper, we prove that: Let G be a connected γt-critical graph of order n (n≥3), then n=Δ(G)(γt(G)−1)+1 if and only if G is regular and, for each v∈V(G), there is an A⊆V(G)−{v} such that N(v)∩A=0?, the subgraph induced by A is 1-regular, and every vertex in V(G)−A−{v} has exactly one neighbor in A. 相似文献
12.
J. Cutler 《Discrete Mathematics》2009,309(9):2749-2754
We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1≤k≤n−2 if G is a connected graph with A⊂V(G) such that dG(v)≥k for all v∈A, then there exists a subtree T of G such that V(T)⊃A and for all v∈A. 相似文献
13.
Kenji Kimura 《Discrete Applied Mathematics》2010,158(18):2071-2074
We extend the notion of a defensive alliance to weighted graphs. Let (G,w) be a weighted graph, where G is a graph and w be a function from V(G) to the set of positive real numbers. A non-empty set of vertices S in G is said to be a weighted defensive alliance if ∑x∈NG(v)∩Sw(x)+w(v)≥∑x∈NG(v)−Sw(x) holds for every vertex v in S. Fricke et al. (2003) [3] have proved that every graph of order n has a defensive alliance of order at most . In this note, we generalize this result to weighted defensive alliances. Let G be a graph of order n. Then we prove that for any weight function w on V(G), (G,w) has a defensive weighted alliance of order at most . We also extend the notion of strong defensive alliance to weighted graphs and generalize a result in Fricke et al. (2003) [3]. 相似文献
14.
A set A⊆V of the vertices of a graph G=(V,E) is an asteroidal set if for each vertex a∈A, the set A\{a} is contained in one component of G−N[a]. The maximum cardinality of an asteroidal set of G, denoted by an (G), is said to be the asteroidal number of G. We investigate structural properties of graphs of bounded asteroidal number. For every k≥1, an (G)≤k if and only if an (H)≤k for every minimal triangulation H of G. A dominating target is a set D of vertices such that D∪S is a dominating set of G for every set S such that G[D∪S] is connected. We show that every graph G has a dominating target with at most an (G) vertices. Finally, a connected graph G has a spanning tree T such that d
T
(x,y)−d
G
(x,y)≤3·|D|−1 for every pair x,y of vertices and every dominating target D of G.
Received: July 3, 1998 Final version received: August 10, 1999 相似文献
15.
Blockers and transversals in some subclasses of bipartite graphs: When caterpillars are dancing on a grid 总被引:3,自引:0,他引:3
B. Ries 《Discrete Mathematics》2010,310(1):132-1946
Given an undirected graph G=(V,E) with matching number ν(G), a d-blocker is a subset of edges B such that ν((V,E?B))≤ν(G)−d and a d-transversal T is a subset of edges such that every maximum matching M has |M∩T|≥d. While the associated decision problem is NP-complete in bipartite graphs we show how to construct efficiently minimum d-transversals and minimum d-blockers in the special cases where G is a grid graph or a tree. 相似文献
16.
Equitable colorings of Kronecker products of graphs 总被引:1,自引:0,他引:1
Wu-Hsiung Lin 《Discrete Applied Mathematics》2010,158(16):1816-1826
For a positive integer k, a graph G is equitably k-colorable if there is a mapping f:V(G)→{1,2,…,k} such that f(x)≠f(y) whenever xy∈E(G) and ||f−1(i)|−|f−1(j)||≤1 for 1≤i<j≤k. The equitable chromatic number of a graph G, denoted by χ=(G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by , is the minimum t such that G is equitably k-colorable for k≥t. The current paper studies equitable chromatic numbers of Kronecker products of graphs. In particular, we give exact values or upper bounds on χ=(G×H) and when G and H are complete graphs, bipartite graphs, paths or cycles. 相似文献
17.
Let G be a graph and f be a mapping from V(G) to the positive integers. A subgraph T of G is called an f‐tree if T forms a tree and dT(x)≤f(x) for any x∈V(T). We propose a conjecture on the existence of a spanning f‐tree, and give a partial solution to it. © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 173–184, 2010 相似文献
18.
Tomoki Yamashita 《Discrete Mathematics》2008,308(9):1620-1627
For a graph G, let σk(G) be the minimum degree sum of an independent set of k vertices. Ore showed that if G is a graph of order n?3 with σ2(G)?n then G is hamiltonian. Let κ(G) be the connectivity of a graph G. Bauer, Broersma, Li and Veldman proved that if G is a 2-connected graph on n vertices with σ3(G)?n+κ(G), then G is hamiltonian. On the other hand, Bondy showed that if G is a 2-connected graph on n vertices with σ3(G)?n+2, then each longest cycle of G is a dominating cycle. In this paper, we prove that if G is a 3-connected graph on n vertices with σ4(G)?n+κ(G)+3, then G contains a longest cycle which is a dominating cycle. 相似文献
19.
Chandan K. Dubey 《Discrete Applied Mathematics》2009,157(1):149-163
An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a1,b1),(a2,b2),…,(am,bm) such that G(V−{aj1,bj1,…,ajk,bjk}) is also an X-critical graph for arbitrary set of indices {j1,…,jk}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x1,y1),…,(xt,yt) such that xi,yi∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G. 相似文献
20.
Let T be any tree of order d≥1. We prove that every connected graph G with minimum degree d contains a subtree T′ isomorphic to T such that G−V(T′) is connected. 相似文献