اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدBook crossing numbers of the complete graph and small local convex crossing numbers
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A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G $, denoted by $ u_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ u_k(K_n) $ for all $ kgeq 14 $ and determine $ u_k(K_n) $ whenever $ 2 < n/k leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ ell $ can have for $ ellleq 4 $.
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Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=si_1si_2 cdotssi_d$ of a Coxeter element $c$ o
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