Ad
related to: chromatic graph chart
Search results
Results from the WOW.Com Content Network
The Grötzsch graph is an example of a 4-chromatic graph without a triangle, and the example can be generalized to the Mycielskians. Theorem (William T. Tutte 1947, [10] Alexander Zykov 1949, Jan Mycielski 1955): There exist triangle-free graphs with arbitrarily high chromatic number.
For a graph G, let χ(G) denote the chromatic number and Δ(G) the maximum degree of G.The list coloring number ch(G) satisfies the following properties.. ch(G) ≥ χ(G).A k-list-colorable graph must in particular have a list coloring when every vertex is assigned the same list of k colors, which corresponds to a usual k-coloring.
A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has a proper coloring. In other words, when the list for each edge has length k , no matter which colors are put in each list, a color can be selected from each list so that G is ...
The Thue number (a variant of the chromatic index) of the Petersen graph is 5. The Petersen graph requires at least three colors in any (possibly improper) coloring that breaks all of its symmetries; that is, its distinguishing number is three. Except for the complete graphs, it is the only Kneser graph whose distinguishing number is not two.
The 7 cycles of the wheel graph W 4. For odd values of n, W n is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. For even n, W n has chromatic number 4, and (when n ≥ 6) is not perfect. W 7 is the only wheel graph that is a unit distance graph in the ...
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable.
No graph can be 0-colored, so 0 is always a chromatic root. Only edgeless graphs can be 1-colored, so 1 is a chromatic root of every graph with at least one edge. On the other hand, except for these two points, no graph can have a chromatic root at a real number smaller than or equal to 32/27. [8]
Therefore, the chromatic number of a subgraph cannot be larger than the chromatic number of the whole graph. The De Bruijn–Erdős theorem concerns the chromatic numbers of infinite graphs, and shows that (again, assuming the axiom of choice) they can be calculated from the chromatic numbers of their finite subgraphs.
Ad
related to: chromatic graph chart