204993527
The size of graphs with clique number m and without nowhere-zero 4-flows
Abstract Let G be a 2-edge-connected simple graph on n ⩾24 vertices containng no K m +1. Of |E(G)|⩾( n - 17 - k 2 )+(m - l)( k + 1 2 )+ 33, where ⌊ n −17/ m ⌋, then either G has a nowhere-zero 4-flow, or G can be contracted to the Petersen graph. This is a generalization of a result in [4].
30786237
Spanning closed trails in graphs
Abstract Let G be a 2-edge-connected simple graph on n >;95 vertices. Let l be the number of vertices of degree 2 in G . We prove that if l n ⧸5−19 and if, for every edge uvϵE ( G ), d ( v )+ d ( v )⩾2 n ⧸5−2, then exactly one of the following holds: (a) G has a spanning closed trail; (b) G can be contracted to K 2, c −2 , where c ⩽ max {5, 3+ l } is an odd number. An example shows that if a graph satisfies the conditions above except that it has too many vertices of degree 2, then the conclusion fails. This result is related to a conjecture of Benhocine et al. (1986), recently proved by Veldman. We obtain some other related results.
11497116
Nowhere zero flows in line graphs
Abstract Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero k-flow would imply that L ( G ), the line graph of G, also has a nowhere zero k-flow. The validity of Tutte's flow conjectures on line graphs would also imply the truth of these conjectures in general.
10941860
Fan-Type Conditions for Collapsible Graphs
A graph G is collapsible if for every even subset R <;; V ( G), there is a spanning connected subgraph of G whose set of odd degree vertices is R. A graph is supereulerian if it contains a spanning closed trail. It is known that every collapsible graph is supereulerian. A graph G of order n is said to satisfy a Fan­ type condition if max{d(u),d(v)} 2: n/(g 2)p E for each pair of vertices u, v at distance two1 where g E {3, 4} is the girth of G, and p 2: 2 and , 2: 0 are fixed numbers. In this paper, we study the Fan-type conditions for collapsible graphs and supereulerian graphs.
17921413
A self-learning worm using importance scanning
The use of side information by an attacker can help a worm speed up the propagation. This philosophy has been the basis for advanced worm scanning mechanisms such as hitlist scanning, routable scanning, and importance scanning. Some of these scanning methods use information on vulnerable hosts. Such information, however, may not be easy to collect before a worm is released. Questions then arise whether and how a worm can self-learn and use such information while propagating, and how virulent the resulting worm may be. In this paper, we design a self-learning worm using importance scanning. An optimal yet practical importance-scanning strategy is derived based on a new metric. A self-learning worm is demonstrated to have the ability to accurately estimate the underlying vulnerable-host distribution if a sufficient number of infected hosts are observed. Experimental results based on parameters chosen from Code Red show that after accurately estimating the distribution of vulnerable hosts, a self-learning worm can spread much faster than a random-scanning worm, a permutation-scanning worm, and a Class A routing worm. Some guidelines for detecting and defending against such self learning worms are also discussed.
207130305
On k-pairable graphs
Motivated by a 1994 result of Graham et al. (Amer. Math. Monthly 101(7) (1994) 664) about spanning trees of the graphs with an antipodal isomorphism, we introduce the concept of k-pairable graphs and extend the result in Graham et al. (Amer. Math. Monthly 101(7) (1994) 664) to this larger class of graphs. We then define a new graph parameter p(G), called the pair length of graph G. This parameter measures the maximum distance, in some sense, between a subgraph induced by half the vertices of G with the isomorphic subgraph induced by the other half of V(G). An upper bound for the parameter p(G) is given. Some properties of the k-pairable graphs and their product graphs are studied. We also post some problems for further research.
40322419
A Quadratic Lower Bound for Rocchio's Similarity-Based Relevance Feedback Algorithm
It is shown in [4] that Rocchio's similarity-based relevance feedback algorithm makes Ωn mistakes in searching for a collection of documents represented by a monotone disjunction of at most k relevant features or terms over the n-dimensional binary vector space {0, 1}n. In practice, Rocchio's algorithm often uses a fixed query updating factor and a fixed classification threshold. When this is the case, we strengthen the work in [4] in this paper and prove that Rocchio's algorithm makes Ωkn---k mistakes in searching for the same collection of documents over the binary vector space {0, 1}n. A quadratic lower bound is obtained when k is proportional to n. An Okn---k2 upper bound is also obtained.
29625123
A PTAS for a disc covering problem using width-bounded separators
In this paper, we study the following disc covering problem: Given a set of discs of various radii on the plane and centers on the grid points, find a subset of discs to maximize the area covered by exactly one disc. This problem originates from the application in digital halftoning, with the best known approximation factor being 5.83 (Asano et al., 2004). We show that if their radii are between two positive constants, then there exists a polynomial time approximation scheme. Our techniques are based on the width-bounded geometric separator recently developed in Fu and Wang (2004), Fu (2006).
701
Common-Face Embeddings of Planar Graphs
Given a planar graph $\Ggg$ and a sequence ${\CC}_1,\ldots,{\CC}_q$, where each ${\CC}_i$ is a family of vertex subsets of $\Ggg$, we wish to find a plane embedding of $\Ggg$, if any exists, such that, for each $i\in\{1,\ldots,q\}$, there is a face Fi in the embedding whose boundary contains at least one vertex from each set in CCi. This problem has applications in the recovery of topological information from geographical data and the design of constrained layouts in VLSI. Let $\inputsize$ be the input size, i.e., the total number of vertices and edges in $\Ggg$ and the families CCi, counting multiplicity. We show that this problem is NP-complete in general. We also show that it is solvable in $O(\inputsize\log \inputsize)$ time for the special case in which, for each input family CCi, each set in CCi induces a connected subgraph of the input graph $\Ggg$. Note that the classical problem of simply finding a planar embedding is a further special case of this case with q=0. Therefore, the processing of the additional constraints CC1, . . .,CCq incurs only a logarithmic factor of overhead.