Search results
Results from the WOW.Com Content Network
Geometric discrepancy theory [1] is a sub-field of discrepancy theory, that deals with balancing geometric sets, such as intervals or rectangles.The general research question in this field is: given a set of points in a geometric space, and a set of objects in the same space, can we color each point in one of two different colors (e.g. black and white), such that each object contains roughly ...
According to Jensen & Toft (1995), the problem was first formulated by Nelson in 1950, and first published by Gardner (1960). Hadwiger (1945) had earlier published a related result, showing that any cover of the plane by five congruent closed sets contains a unit distance in one of the sets, and he also mentioned the problem in a later paper (Hadwiger 1961).
The case = is also easy: the graphs requiring three colors are the non-bipartite graphs, and every non-bipartite graph has an odd cycle, which can be contracted to a 3-cycle, that is, a minor. In the same paper in which he introduced the conjecture, Hadwiger proved its truth for k = 4 {\displaystyle k=4} .
An edge coloring with k colors is called a k-edge-coloring and is equivalent to the problem of partitioning the edge set into k matchings. The smallest number of colors needed for an edge coloring of a graph G is the chromatic index, or edge chromatic number, χ ′ (G). A Tait coloring is a 3-edge coloring of a cubic graph.
Domain coloring plot of the function f(x) = (x 2 − 1)(x − 2 − i) 2 / x 2 + 2 + 2i , using the structured color function described below. In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane. By assigning points on the ...
A 3-edge-coloring of the Desargues graph. In graph theory, a proper edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green.
In 1970, a geometric coloring problem equivalent to the Mirsky–Newman theorem was given in the Soviet mathematical olympiad: suppose that the vertices of a regular polygon are colored in such a way that every color class itself forms the vertices of a regular polygon. Then, there exist two color classes that form congruent polygons.
Graph theory is the branch of mathematics that examines the properties of mathematical graphs.See glossary of graph theory for common terms and their definition.. Informally, this type of graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs), which can also have associated directions.