Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph. There are many generalizations of classic hypergraph coloring.
Classic coloring theory is the theory for finding the minimum number of colors.
Hypergraph coloring number.
Hypergraphs with both edges and co edges are called mixed hypergraphs.
And the hypergraph dual of vertex coloring is edge coloring.
The wider structure hypergraphs.
We prove that there is a constant cdepending only on ksuch that every simple k uniform hypergraph hwith maximum degree has chromatic number satisfying h c log 1 k 1.
Hypergraph coloring is a well studied problem in the liter ature in discrete mathematics combinatorics and computer science.
Thus the erdosfaberlovasz conjecture is equivalent to the statement that any simple hypergraph with n vertices has chromatic index edge coloring number at most n.
In contrast to graphs where one can decide in linear time if a graph is 2 colorable or equiv alently bipartite testing if a given hypergraph is 2 colorable is np hard even.
Coloring simple hypergraphs alan frieze dhruv mubayiy october 1 2013 abstract fix an integer k 3.
In this paper we introduce and study a generalization of hypergraph coloring which we call hypergraph dp coloring.
A k uniform hypergraph is simple if every two edges share at most one vertex.
The maximal number of colors for which there exists a mixed hypergraph coloring using all the colors is called the upper chromatic number of a hypergraph h and is denoted by h.
We introduce the notion of a co edge of a hypergraph which is a subset of vertices to be colored so that at least two vertices are of the same color.
The 2 colorable hypergraphs are exactly the bipartite ones.
The aim of this section is to investigate the homotopy type of the coloring complex of an arbitrary hypergraph.
Pdf we discuss the colouring theory of finite set systems.
Whereas classical coloring complexes of graphs are known to be homotopy equivalent to wedges of spheres of top dimension it turns out that for hypergraph coloring complexes not that much can be said.
Hypergraphs for which there exists a coloring using up to k colors are referred to as k colorable.
This is not merely an extension of results from collections of 2 element sets graphs to larger sets.
The maximal number of colors.
The basic idea of mixed hypergraphs is to introduce the problem of finding the maximum number of colors in the most general setting and mix it with the old approach.
This is an extension of the similar concept for graphs ie 2 uniform hypergraphs that was developed by dvorak and postle under the name correspondence coloring.