38400231
On mixed Ramsey numbers
Abstract For a graph theoretic parameter f , an integer m and a graph H , the mixed Ramsey number v(f; m; H) is defined as the least positive integer p such that if G is any graph of order p , then either f(G) ⩾ m or Ḡ contains a subgraph isomorphic to H . Let ϱ denote vertex linear arboricity and let H be any connected graph of order n . In this note we show that v(ϱ; m; H) = 1 + (n + n p ( H ) − 2)(m − 1) , where n p ( H ) is the path partition number of H .
24660543
A note on the lower bounds of signed domination number of a graph
Abstract Let G = (V, E) be a graph. For a function f : V → {−1, 1}, the weight of f is w(f) = Σv ∈ V f(v). For a vertex v in V, we define f[v] = Σu∈N[v] f(u). A signed dominating function of G is a function f : V → {−1, 1} such that f[v] ⩾ 1 for all v ∈ V. The signed domination number γs(G) of G is the minimum weight of a signed dominating function on G. A signed dominating function of a weight γs(G) we call a γs-function of G. In this paper, we study the signed domination problem of general graph, and obtain some lower bounds of the signed domination number of a graph, and show that these lower bounds are sharp, and extend a result in Dunbar et al. (1995).
31788400
Extremal graphs for inequalities involving domination parameters
Abstract A characterization of n-vertex isolate-free connected graphs G whose domination number γ ( G ) satisfies γ(G)=⌊n/2⌋ is obtained. This result enables us to obtain extremal graphs of inequalities which bound the sum of two domination parameters in isolate-free graphs.
6156170
On edge domination numbers of graphs
Let @c"s^'(G) and @c"s"s^'(G) be the signed edge domination number and signed star domination number of G, respectively. We prove that 2n-4>[email protected]"s"s^'(G)>[email protected]"s^'(G)>=n-m holds for all graphs G without isolated vertices, where n=|V(G)|>=4 and m=|E(G)|, and pose some problems and conjectures.
39744832
On signed edge domination numbers of graphs
Abstract Given a graph G=(V,E) , if e=uv∈E , then the closed edge-neighbourhood of e is denoted by N[e]={u′v′∈E|u′=u or v′=v} . A function f : E→{+1,−1} is called the signed edge domination function (SEDF) of G if ∑ e′∈N[e] f(e′)⩾1 for every e∈E . The signed edge domination number γ s ′(G) of G is defined as γ s ′(G)= min {∑ e∈E f(e) | f is an SEDF of G} . Let Ψ(m)= min {γ s ′(H)|H is a graph with |E(H)|=m} . In this paper, we determine the exact value of Ψ(m) for each positive integer m . That is: Ψ(m)=2 1 3 24m+25 +6m+5 6 −m, where ⌈x⌉ denotes the ceiling of x . In addition, we also characterize all connected graphs G with γ s ′(G)=|E(G)| .
29802636
Two classes of edge domination in graphs
Let γ's(G) (γ'l(G), resp.) be the number of (local) signed edge domination of a graph G [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179-189]. In this paper, we prove mainly that γ's(G) ≤ ⌊11/6n - 1⌋ and γ's(G) ≤ 2n - 4 hold for any graph G of order n (n ≥ 4), and pose several open problems and conjectures.
31813405
Website evolution based on statistic data
Maintenance and evolution are critical for website since the requirements often change, the developing cycle is short, while the life cycle is long. In order to make progress in this area, we focus on the users' responses and attitudes. So we begin our work at the log files on the side of website server, gathering the users' visiting information and the server's responses. Thus we obtain our weighted structure model. Then, we analyze and discuss such information to identify the key pages, the predominate pages and the users' visiting patterns. And we can improve the testing efficiency based on these knowledge. All these will help to improve the structure of the site, fulfill the functionality of the site, and enhance users' visiting efficiency.
33924916
Geometric Algorithms for Density-Based Data Clustering
We present new geometric approximation and exact algorithms for the density-based data clustering problem in d-dimensional space Rd (for any constant integer d ? 2). Previously known algorithms for this problem are efficient only for uniformly-distributed points. However, these algorithms all run in ?(n2) time in the worst case, where n is the number of input points. Our approximation algorithm based on the ?-fuzzy distance function takes O(n log n) time for any given fixed value ? > 0, and our exact algorithms take sub-quadratic time. The running times and output quality of our algorithms do not depend on any particular data distribution. We believe that our fast approximation algorithm is of considerable practical importance, while our sub-quadratic exact algorithms are more of theoretical interest. We implemented our approximation algorithm and the experimental results show that our approximation algorithm is efficient on arbitrary input point sets.
20988039
Two conjectures equivalent to the perfect graph conjecture
Abstract In this note, we study the graphs with the property that each of their induced subgraphs has circular clique number (defined in Section 2) the same as its clique number, prove that a graph satisfies such a property if and only if neither itself nor its complement contains an induced subgraph isomorphic to an odd cycle of length at least five, and then give two conjectures which are equivalent to the perfect graph conjecture.
16379968
On Structure of Some Plane Graphs with Application to Choosability
A graph G=(V, E) is (x, y)-choosable for integers xy?1 if for any given family {A(v)?v?V} of sets A(v) of cardinality x, there exists a collection {B(v)?v?V} of subsets B(v)?A(v) of cardinality y such that B(u)?B(v)=? whenever uv?E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k?{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable.
5196479
Odd Wheels in Graphs
For k?1 the odd wheel of 2k+1 spokes, denoted by W2k+1, is the graph obtained from a cycle of length 2k+1 by adding a new vertex and joining it to all vertices of the cycle. In this paper it is shown that if a graph G of order n with minimum degree greater than 7n/12 is at least 4-chromatic then G contains an odd wheel with at most 5 spokes.