共查询到20条相似文献,搜索用时 15 毫秒
1.
Xiaoling Zhang 《Linear algebra and its applications》2008,428(7):1610-1619
In this paper, we study the largest Laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs, respectively. Identifying the center of a star K1,k and one vertex of degree n of Km,n, we denote by the resulting graph. We show that the graph (1?k?n-4) is the unique graph with the largest Laplacian spectral radius among the bipartite graphs with n vertices and k cut edges, and (n?7) is the unique graph with the largest Laplacian spectral radius among all the bicyclic bipartite graphs. 相似文献
2.
A Roman domination function on a graph G=(V(G),E(G)) is a function f:V(G)→{0,1,2} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex v for which f(v)=2. The weight of a Roman dominating function is the value f(V(G))=∑u∈V(G)f(u). The minimum weight of a Roman dominating function on a graph G is called the Roman domination number of G. Cockayne et al. [E. J. Cockayne et al. Roman domination in graphs, Discrete Mathematics 278 (2004) 11-22] showed that γ(G)≤γR(G)≤2γ(G) and defined a graph G to be Roman if γR(G)=2γ(G). In this article, the authors gave several classes of Roman graphs: P3k,P3k+2,C3k,C3k+2 for k≥1, Km,n for min{m,n}≠2, and any graph G with γ(G)=1; In this paper, we research on regular Roman graphs and prove that: (1) the circulant graphs and , n⁄≡1 (mod (2k+1)), (n≠2k) are Roman graphs, (2) the generalized Petersen graphs P(n,2k+1)( (mod 4) and ), P(n,1) (n⁄≡2 (mod 4)), P(n,3) ( (mod 4)) and P(11,3) are Roman graphs, and (3) the Cartesian product graphs are Roman graphs. 相似文献
3.
Evgenii L. Bashkirov 《Journal of Pure and Applied Algebra》2010,214(7):1049-1062
Let a field K be an algebraic extension of a subfield k of characteristic not 2, n an integer, a non-degenerate isotropic form in n variables over K with coefficients in k. We study subgroups of the orthogonal group On(K,Q) that contain the derived subgroup Ωn(k,Q) of the group On(k,Q). 相似文献
4.
For given graphs G and H, the Ramsey numberR(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper, we investigate the Ramsey number R(∪G,H), where G is a tree and H is a wheel Wm or a complete graph Km. We show that if n?3, then R(kSn,W4)=(k+1)n for k?2, even n and R(kSn,W4)=(k+1)n-1 for k?1 and odd n. We also show that . 相似文献
5.
For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number of a disjoint union of graphs . For any natural integer k, we contain a general upper bound, R(kG,H)?R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4, 2n-8 or 2n-6, then R(kSn,Wm)=R(Sn,Wm)+(k-1)n. Furthermore, if |Gi|>(|Gi|-|Gi+1|)(χ(H)-1) and R(Gi,H)=(χ(H)-1)(|Gi|-1)+1, for each i, then . 相似文献
6.
For given graphs G1,G2,…,Gk, k≥2, the multicolor Ramsey number, denoted by R(G1,G2,…,Gk), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of Gi colored with i, for some 1≤i≤k. Let Pk (resp. Ck) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(Pi,Pk,Cm) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P3,Pk,C4) for all integers k≥3. 相似文献
7.
Fu-Tao Hu 《Discrete Mathematics》2010,310(4):877-886
Let n and k be integers with n≥k≥0. This paper presents a new class of graphs H(n,k), which contains hypercubes and some well-known graphs, such as Johnson graphs, Kneser graphs and Petersen graph, as its subgraphs. The authors present some results of algebraic and topological properties of H(n,k). For example, H(n,k) is a Cayley graph, the automorphism group of H(n,k) contains a subgroup of order 2nn! and H(n,k) has a maximal connectivity and is hamiltonian if k is odd; it consists of two isomorphic connected components if k is even. Moreover, the diameter of H(n,k) is determined if k is odd. 相似文献
8.
Bing Li 《Journal of Mathematical Analysis and Applications》2008,339(2):1322-1331
For any real number β>1, let ε(1,β)=(ε1(1),ε2(1),…,εn(1),…) be the infinite β-expansion of 1. Define . Let x∈[0,1) be an irrational number. We denote by kn(x) the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x. If is bounded, we obtain that for all x∈[0,1)?Q,
9.
Michael Cavers 《Discrete Mathematics》2008,308(10):2011-2017
In this paper, we prove that for any forest F⊂Kn, the edges of E(Kn)?E(F) can be partitioned into O(nlogn) cliques. This extends earlier results on clique partitions of the complement of a perfect matching and of a hamiltonian path in Kn.In the second part of the paper, we show that for n sufficiently large and any ε∈(0,1], if a graph G has maximum degree O(n1-ε), then the edges of E(Kn)?E(G) can be partitioned into cliques provided there exist certain Steiner systems. Furthermore, we show that there are such graphs G for which Ω(ε2n2-2ε) cliques are required in every clique partition of E(Kn)?E(G). 相似文献
10.
For a given graph H and a positive n, the rainbow number ofH, denoted by rb(n,H), is the minimum integer k so that in any edge-coloring of Kn with k colors there is a copy of H whose edges have distinct colors. In 2004, Schiermeyer determined rb(n,kK2) for all n≥3k+3. The case for smaller values of n (namely, ) remained generally open. In this paper we extend Schiermeyer’s result to all plausible n and hence determine the rainbow number of matchings. 相似文献
11.
12.
Lingsheng Shi 《Discrete Mathematics》2007,307(2):290-292
The Ramsey number R(G) of a graph G is the least integer p such that for all bicolorings of the edges of the complete graph Kp, one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U,V;E) of order n where the maximum degree of vertices in U is at most , . Moreover, we show that the Ramsey number of the cube Qn of dimension n satisfies . In both cases, the small terms are removed from the powers in the upper bounds of a earlier result of the author. 相似文献
13.
Dhruv Mubayi 《Journal of Combinatorial Theory, Series A》2005,111(1):106-110
Given positive integers n,k,t, with 2?k?n, and t<2k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T?r(n,k,t) and Tr(n,k,t), where T?r(n,k,t) (Tr(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T?r(n,k,t)=(1+o(1))Tr(n,k,t) for fixed r,k,t with and n→∞. 相似文献
14.
Edith Hemaspaandra Lane A. Hemaspaandra Stanis?aw P. Radziszowski 《Discrete Applied Mathematics》2007,155(2):103-118
We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:
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- , , , and .
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- For all k?2, and .
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- For all k?2, .
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- .
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- For all k?2, .
15.
Y. Caro 《Discrete Mathematics》2010,310(4):742-747
For a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already , where n is the number of vertices of G. It is conjectured that for every fixed k. We prove this for k=2,3. While the proof for the case k=2 is easy, already the proof for the case k=3 is considerably more difficult. The case k=4 remains open.A related parameter, sk(G), denotes the maximum integer m so that there are k vertex-disjoint subgraphs of G, each with m vertices, and with the same maximum degree. We prove that for every fixed k, sk(G)≥n/k−o(n). The proof relies on probabilistic arguments. 相似文献
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17.
Oleg Pikhurko 《Discrete Mathematics》2006,306(17):2097-2107
A graph G of order n and size m is edge-magic if there is a bijection l:V(G)∪E(G)→[n+m] such that all sums l(a)+l(b)+l(ab), ab∈E(G), are the same. We present new lower and upper bounds on M(n), the maximum size of an edge-magic graph of order n, being the first to show an upper bound of the form . Concrete estimates for ε can be obtained by knowing s(k,n), the maximum number of distinct pairwise sums that a k-subset of [n] can have.So, we also study s(k,n), motivated by the above connections to edge-magic graphs and by the fact that a few known functions from additive number theory can be expressed via s(k,n). For example, our estimate
18.
K.L. Ng 《Discrete Mathematics》2009,309(6):1603-1610
For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs Kp and Kq by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4. 相似文献
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20.
Gábor N. Sárközy 《Discrete Mathematics》2009,309(6):1611-1622
Suppose that 0<η<1 is given. We call a graph, G, on n vertices an η-Chvátal graph if its degree sequence d1≤d2≤?≤dn satisfies: for k<n/2, dk≤min{k+ηn,n/2} implies dn−k−ηn≥n−k. (Thus for η=0 we get the well-known Chvátal graphs.) An -algorithm is presented which accepts as input an η-Chvátal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best -algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs (δ(G)≥n/2 where δ(G) is the minimum degree in G). 相似文献