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K-tree
About: K-tree is a research topic. Over the lifetime, 427 publications have been published within this topic receiving 12096 citations.
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TL;DR: This work considers several random graph models based on k‐trees, which can be generated by applying the probabilistic growth rules “uniform attachment’, “preferential attachment”, or a “saturation”‐rule, respectively, but which also can be described in a combinatorial way.
Abstract: We consider several random graph models based on k-trees, which can be generated by applying the probabilistic growth rules "uniform attachment", "preferential attachment", or a "saturation"-rule, respectively, but which also can be described in a combinatorial way. For all of these models we study the number of ancestors and the number of descendants of nodes in the graph by carrying out a precise analysis which leads to exact and limiting distributional results. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 465-489, 2014
13 citations
TL;DR: It is shown that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.
Abstract: Generalized alcoved polytopes are polytopes whose facet normals are roots in a given root system. We call a set of points in an alcoved polytope a generating set if there does not exist a strictly smaller alcoved polytope containing it. The type $A$ alcoved polytopes are precisely the tropical polytopes that are also convex in the usual sense. In this case the tropical generators form a generating set. We show that for any root system other than $F_4$, every alcoved polytope invariant under the natural Weyl group action has a generating set of cardinality equal to the Coxeter number of the root system.
13 citations
TL;DR: This note addresses the following question: Which graphs G on n vertices with w ( G ) = r have the maximum number of cliques?
13 citations
TL;DR: It is proved that the local clique cover number of every claw-free graph is at most cΔ/logΔ, where Δ is the maximum degree of the graph and c is a constant.
Abstract: A k-clique covering of a simple graph G is a collection of cliques of G covering all the edges of G such that each vertex is contained in at most k cliques. The smallest k for which G admits a k-clique covering is called the local clique cover number of G and is denoted by lccG. Local clique cover number can be viewed as the local counterpart of the clique cover number that is equal to the minimum total number of cliques covering all edges. In this article, several aspects of the local clique covering problem are studied and its relationships to other well-known problems are discussed. In particular, it is proved that the local clique cover number of every claw-free graph is at most cΔ/logΔ, where Δ is the maximum degree of the graph and c is a constant. It is also shown that the bound is tight, up to a constant factor. Moreover, regarding a conjecture by Chen eti¾?al. Clique covering the edges of a locally cobipartite graph, Discrete Math 2191-32000, 17-26, we prove that the clique cover number of every connected claw-free graph on n vertices with the minimum degree i¾?, is at most n+ci¾?4/3log1/3i¾?, where c is a constant.
13 citations
TL;DR: It is shown that if T(G)>=@t(G-S)@w( G-S)+1k-2, for any subset S of V(G), with k>=3, then G has a k-tree.
13 citations