We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degree, except when simple arithmetic forces the edge-chromatic number to be one greater than the maximum degree. For Mycielski graphs, we strengthen an old result that the graphs are Hamiltonian by showing that they are Hamilton-connected (except M 3 , which is a cycle). For Keller graphs G d , we establish, in all cases, the exact value of the chromatic number, the edge-chromatic number, and the independence number; and we get the clique covering number in all cases except 5 ≤ d ≤ 7. We also investigate Hamiltonian decompositions of Keller graphs, obtaining them up to G 6 .
Abstract. It is proved that if F is a convex closed set in C n , n ≥ 2, containing at most one (n − 1)-dimensional complex hyperplane, then the Kobayashi metric and the Lempert function of C n \ F identically vanish.Let D be a domain in These invariants can be characterized as the largest metric and function which decrease under holomorphic mappings and coincide with the Poincaré metric and distance on ∆. It is well known that if D is a bounded domain in C n , or a plane domain whose complement contains at least two points, then K D (z, X) > 0 for X = 0 and D (z, w) > 0 for z = w. On the other hand, the Kobayashi metric and the Lempert function of a plane domain whose complement contains at most one point identically vanish. Note also that there are domains in C n with bounded connected complements and non-vanishing Kobayashi metrics and Lempert functions. For example, if z 0 is a strictly pseudoconvex boundary point of a domain D in C n , n ≥ 2, then (cf.
Abstract. For a domain D ⊂ C the Kobayashi-Royden κ and Hahn h pseudometrics are equal iff D is simply connected. Overholt showed that for D ⊂ C n , n ≥ 3, we haveThe aim of this paper is to show that h D1×D2 ≡ κ D1×D2 iff at least one of D 1 , D 2 is simply connected or biholomorphic to C \ {0}. In particular, there are domains D ⊂ C 2 for which h D ≡ κ D .
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