Star Colouring In Graph Theory. Star arboricity is the minimum number of forests that a graph can be partitioned into such that each tree in each forest is a star and the star chromatic number of a graph is the minimum number of colors needed to color its vertices in such a way that every two color classes together form a subgraph in which all connected components are stars. A star coloring of an undirected graph G is a proper vertex coloring of G ie no two neighbors are assigned the same color such that any path of length 3 in G is not bicolored.
Star arboricity is the minimum number of forests that a graph can be partitioned into such that each tree in each forest is a star and the star chromatic number of a graph is the minimum number of colors needed to color its vertices in such a way that every two color classes together form a subgraph in which all connected components are stars. This problem is an outgrowth of the well-known four-colour map problem which asks whether the countries on every map can be coloured by using just four colours in such a way that countries sharing an edge have different colours. Equivalently in a star coloring the induced subgraphs.
Simply put no two vertices of an edge should be of the same color.
Star arboricity is the minimum number of forests that a graph can be partitioned into such that each tree in each forest is a star and the star chromatic number of a graph is the minimum number of colors needed to color its vertices in such a way that every two color classes together form a subgraph in which all connected components are stars. A proper coloring of the vertices of a graph is called a st r coloringif the union of every two color classes induce a star forest. The star chromatic number of G χsG is the minimum number of colors needed to star color G. A brief review of graph coloring methods in Polish was given by Kubale in 32 and a more detailed one in a book by the same author.
