Hypergraph Coloring Number
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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 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. A k uniform hypergraph is simple if every two edges share at most one vertex.
Coloring simple hypergraphs alan frieze dhruv mubayiy october 1 2013 abstract fix an integer k 3.
Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph. Hypergraphs for which there exists a coloring using up to k colors are referred to as k colorable. The 2 colorable hypergraphs are exactly the bipartite ones. There are many generalizations of classic hypergraph coloring.
We present a hypergraph coloring algorithm and analyze its performance in spaces of random hypergrpahs. In these spaces the number of colors used by our algorithm is almost surely within a small constant factor less than 2 of the weak chromatic number of the hypergraph. This also establishes new upper and lower bounds for the weak chromatic number of uniform hypergraphs. Mixed hypergraph coloring classic coloring theory is the theory for finding the minimum 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. May connect any number of vertices while an edge in a graph can connect only two vertices. A r uniform hypergraphis a hypergraph such that all its hyperedges have size r. A bicoloring of a hypergraph g is a coloring of the vertices of g with two colors.
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Each of the vertices is assigned one of the two colors. Hypergraph ramsey numbers david conlon jacob foxy benny sudakovz abstract the ramsey number r k s n is the minimum nsuch that every red blue coloring of the k tuples of an n element set contains a red set of size sor a blue set of size n where a set is called red. Poulomi ghosh and anita pal 42 iii good coloring let be a hypergraph a good k coloring is a partition i i ij of v such that every hyperedge e contains the largest possible number of different colors i e. Hence the chromatic number based on strong coloring of a cycle in a cluster hypergraph is size of an edge containing the maximum number of maximal nodes.K 2. As the cluster hypergraph contains edges of size two for odd length cycles the chromatic number is three and for even length cycles the chromatic number is two. Np hard to color a 3 uniform 2 colorable hypergraph with 3 colors. For 4 uniform hypergraphs and upwards guruswami h astad and sudan 14 were able to show such a separation i e they showed that it is np hard to color a 2 colorable 4 uniform hypergraph with any constant number of colors this result immediately extends to any k 4.
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