منابع مشابه
Monochromatic triangles in three-coloured graphs
In 1959, Goodman [9] determined the minimum number of monochromatic triangles in a complete graph whose edge set is 2-coloured. Goodman [10] also raised the question of proving analogous results for complete graphs whose edge sets are coloured with more than two colours. In this paper, for n sufficiently large, we determine the minimum number of monochromatic triangles in a 3-coloured copy of K...
متن کاملEmpty Monochromatic Triangles
We consider a variation of a problem stated by Erdös and Guy in 1973 about the number of convex k-gons determined by any set S of n points in the plane. In our setting the points of S are colored and we say that a spanned polygon is monochromatic if all its points are colored with the same color. As a main result we show that any bi-colored set of n points in R in general position determines a ...
متن کاملMonochromatic triangles in two-colored plane
We prove that for any partition of the plane into a closed set C and an open set O and for any configuration T of three points, there is a translated and rotated copy of T contained in C or in O. Apart from that, we consider partitions of the plane into two sets whose common boundary is a union of piecewise linear curves. We show that for any such partition and any configuration T which is a ve...
متن کاملOn vertex coloring without monochromatic triangles
We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of classic and parametrized algorithms. Several computational complexity results are also presented, which improve on the previous results found in the literatur...
متن کاملColouring Planar Graphs With Three Colours and No Large Monochromatic Components
We prove the existence of a function f : N → N such that the vertices of every planar graph with maximum degree ∆ can be 3colored in such a way that each monochromatic component has at most f(∆) vertices. This is best possible (the number of colors cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan, and Ven...
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ژورنال
عنوان ژورنال: Bulletin of the Australian Mathematical Society
سال: 1988
ISSN: 0004-9727,1755-1633
DOI: 10.1017/s0004972700026733