Abstract:We introduce the circular chromatic number c of a digraph and establish various basic results. They show that the coloring theory for digraphs is similar to the coloring theory for undirected graphs when independent sets of vertices are replaced by acyclic sets. Since the directed k-cycle has circular chromatic number k=(k À 1), for k ! 2, values of c between 1 and 2 are possible. We show that in fact, c takes on all rational values greater than 1. Furthermore, there exist digraphs of arbitrarily large digirth and circular chromatic number. It is NP-complete to decide if a given digraph has c at most 2. ß
The minor crossing number of a graph G is defined as the minimum crossing number of all graphs that contain G as a minor. Basic properties of this new invariant are presented. We study topological structure of graphs with bounded minor crossing number and obtain a new strong version of a lower bound based on the genus. We also give a generalization of an inequality of Moreno and Salazar crossing numbers of a graph and its minors.
Abstract:Zip product was recently used in a note establishing the crossing number of the Cartesian product K 1,n P m . In this article, we further investigate the relations of this graph operation with the crossing numbers of graphs. First, we use a refining of the embedding method bound for crossing numbers to weaken the connectivity condition under which the crossing number is additive for the zip product. Next, we deduce a general theorem for bounding the crossing numbers of (capped) Cartesian product of graphs with trees, which yields exact results under certain symmetry conditions. We apply this theorem to obtain exact and approximate results on crossing numbers of Cartesian product of various graphs with trees.
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