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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 . The four color theorem is equivalent to the assertion that every planar cubic bridgeless graph admits a Tait coloring.
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.
The graph of the 3-3 duoprism (the line graph of ,) is perfect.Here it is colored with three colors, with one of its 3-vertex maximum cliques highlighted. In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph.
3. The Lovász number or Lovász theta function of a graph is a graph invariant related to the clique number and chromatic number that can be computed in polynomial time by semidefinite programming. Thomsen graph The Thomsen graph is a name for the complete bipartite graph,. topological 1.
A graph is bipartite if and only if it is 2-colorable, (i.e. its chromatic number is less than or equal to 2). [3] A graph is bipartite if and only if every edge belongs to an odd number of bonds, minimal subsets of edges whose removal increases the number of components of the graph. [16]
The smallest number of colors needed in a (proper) edge coloring of a graph G is the chromatic index, or edge chromatic number, χ′(G). The chromatic index is also sometimes written using the notation χ 1 ( G ) ; in this notation, the subscript one indicates that edges are one-dimensional objects.
In graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. [1] A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. [2]
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.