Summary. This is the first of two papers in which we classify the regular projective polyhedra in P 3 with planar faces. Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such polyhedra, and conclude with the case in which the polyhedron is an embedded surface.Mathematics Subject Classification (1991). 52B, 52C, 51M20, 51F15.
We prove several colorful generalizations of classical theorems in discrete geometry. Moreover, the colorful generalization of Kirchberger's theorem gives a generalization of the theorem of Tverberg on non-separated partitions.
A k-graph, H = (V,E), is tight if for every surjective mapping f : V + { 1,. . . , k } there exists an edge a E f such that f l , is injective. Clearly, 2-graphs are tight if and only if they are connected. Bounds for the minimum number p," of edges in a tight k-graph with n vertices are given. We conjecture that p : = [n(n -2)/31 for every n and prove the equality when 2n + 1 is prime. From the examples, minimal embeddings of complete graphs into surfaces follow.
The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.
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