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Polychromatic Colorings of Plane Graphs
, 2007
"... We show that the vertices of any plane graph in which every face is of length at least g can be colored by ⌊(3g − 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. W ..."
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Cited by 3 (0 self)
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We show that the vertices of any plane graph in which every face is of length at least g can be colored by ⌊(3g − 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 whether a plane graph admits a vertex coloring by 3 colors in which all colors appear in every face is N Pcomplete even for graphs in which all faces are of length 3 or 4 only. If all faces are of length 3 this can be decided in polynomial time. The investigation of this problem is motivated by its connection to a variant of the art gallery problem in computational geometry. 1
Polychromatic Colorings of ndimensional GuillotinePartitions
, 2008
"... A strong hyperboxrespecting coloring of an ndimensional hyperbox partition is a coloring of the corners of its hyperboxes with 2 n colors such that any hyperbox has all the colors appearing on its corners. A guillotinepartition is obtained by starting with a single axisparallel hyperbox and re ..."
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A strong hyperboxrespecting coloring of an ndimensional hyperbox partition is a coloring of the corners of its hyperboxes with 2 n colors such that any hyperbox has all the colors appearing on its corners. A guillotinepartition is obtained by starting with a single axisparallel hyperbox and recursively cutting a hyperbox of the partition into two hyperboxes by a hyperplane orthogonal to one of the n axes. We prove that there is a strong hyperboxrespecting coloring of any ndimensional guillotinepartition. This theorem generalizes the result of Horev et al. [8] who proved the 2dimensional case. This problem is a special case of the ndimensional variant of polychromatic colorings. The proof gives an efficient coloring algorithm as well.
Edge Guards for Polyhedra in ThreeSpace
"... It is shown that every polyhedron in R3 with m edges can be guarded with at most 27 32m edge guards. The bound improves to 5 1 ..."
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It is shown that every polyhedron in R3 with m edges can be guarded with at most 27 32m edge guards. The bound improves to 5 1