Robert Berke

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We show that the vertices of any plane graph in which every face is of size at least <i>g</i> can be colored by (3g &#192;&#253; 5)=4 colors so that every color appears in every face. This is nearly tight, as there are plane graphs that admit no vertex coloring of this type with more than (3g+1)=4 colors. We further show that the problem of determining(More)
We show that any graph of maximum degree at most ¢ has a two-coloring, such that one color-class is an independent set while the other color induces monochromatic components of order at most £ ¥ ¤ § ¦. On the other hand for any constant¨we exhibit a ©-regular graph, such that the deletion of any independent set leaves at least one component of order greater(More)
We study relaxations of proper two-colorings, such that the order of the induced monochro-matic components in one (or both) of the color classes is bounded by a constant. A coloring of a graph G is called (C 1 , C 2)-relaxed if every monochromatic component induced by vertices of the first (second) color is of order at most C 1 (C 2 , resp.). We prove that(More)
Transversals in r-partite graphs with various properties are known to have many applications in graph theory and theoretical computer science. We investigate f-bounded transversals (or f-BT), that is, transversals whose connected components have order at most f. In some sense we search for the the sparsest f-BT-free graphs. We obtain estimates on the(More)
Chautauqua County, known as the " western gateway " to New York State, is a rural county located in the southwest corner of the state. Bordered by Pennsyl-vania to the south and by Lake Erie to the northwest, Chautauqua County has a population of approximately 137,267. The population is distributed across two cities (Jamestown and Dunkirk), 27 towns, and 15(More)
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