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.
5754331
Web-page summarization using clickthrough data
Most previous Web-page summarization methods treat a Web page as plain text. However, such methods fail to uncover the full knowledge associated with a Web page needed in building a high-quality summary, because many of these methods do not consider the hidden relationships in the Web. Uncovering the hidden knowledge is important in building good Web-page summarizers. In this paper, we extract the extra knowledge from the clickthrough data of a Web search engine to improve Web-page summarization. Wefirst analyze the feasibility in utilizing the clickthrough data to enhance Web-page summarization and then propose two adapted summarization methods that take advantage of the relationships discovered from the clickthrough data. For those pages that are not covered by the clickthrough data, we design a thematic lexicon approach to generate implicit knowledge for them. Our methods are evaluated on a dataset consisting of manually annotated pages as well as a large dataset that is crawled from the Open Directory Project website. The experimental results indicate that significant improvements can be achieved through our proposed summarizer as compared to the summarizers that do not use the clickthrough data.
9623784
Multi-type Features Based Web Document Clustering
Clustering has been demonstrated as a feasible way to explore the contents of document collection and organize search engine results. For this task, many features of Web page, such as content, anchor text, URL, hyperlink etc, can be exploited and different results can be obtained. We expect to provide a unified and even better result for end users. Some work have studied how to use several types of features together to perform clustering. Most of them focus on ensemble method or combination of similarity. In this paper, we propose a novel algorithm: Multi-type Features based Reinforcement Clustering (MFRC). This algorithm does not use a unique combine score for all feature spaces, but uses the intermediate clustering result in one feature space as additional information to gradually enhance clustering in other spaces. Finally a consensus can be achieved by such mutual reinforcement. And the experimental results show that MFRC also provides some performance improvement.
28886116
A Linear-Time Algorithm for 7-Coloring 1-Plane Graphs
AbstractA graph G is 1-planar if it can be embedded in the plane in such a way 
that each edge crosses at most one other edge. Borodin showed that 
1-planar graphs are 6-colorable, but his proof does not lead to a linear-time 
algorithm. This paper presents a linear-time algorithm for 7-coloring 
1-plane graphs (which are 1-planar graphs already embedded in the plane). 
The main difficulty in the design of our algorithm comes from 
the fact that the class of 1-planar graphs is not closed under the 
operation of edge contraction. This difficulty is overcome by a structural 
lemma that may be useful in other problems on 1-planar graphs. This paper 
also shows that it is NP-complete to decide whether a given 1-planar graph 
is 4-colorable. The complexity of the problem of deciding whether a given 
1-planar graph is 5-colorable is still unknown.
43148591
Strategies for Personalized Search Engines
1865477
On integral sum graphs
A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v)=f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K"3 has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous related results are also presented.