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Person# Jan Kyncl

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In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called no

Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal

An ordered graph is a pair G = (G,

We generalize the ham sandwich theorem to d +1 measures on R-d as follows. Let mu(1), mu(2),..., mu(d+1) be absolutely continuous finite Borel measures on R-d. Let omega(i) = mu(i) (R-d) for i is an element of [d + 1], omega = min{omega(i) : i is an element of[d + 1]) and assume that Sigma(d+1)(j=1) omega(j)= 1. Assume that omega(i) = 1/(d + 1). As a consequence we obtain that every (d + 1)-colored set of nd points in R-d such that no color is used for more than n points can be partitioned into n disjoint rainbow (d - 1)-dimensional simplices. (c) 2017 Elsevier B.V. All rights reserved.

An abstract topological graph (briefly an AT-graph) is a pair A = (G, X) where G = (V, E) is a graph and X. E2 is a set of pairs of its edges. The AT-graph A is simply realizable if G can be drawn in the plane so that each pair of edges from X crosses exactly once and no other pair crosses. We showthat simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent Z2-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.