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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.
The choosability (or list colorability or list chromatic number) ch(G) of a graph G is the least number k such that G is k-choosable. More generally, for a function f assigning a positive integer f(v) to each vertex v, a graph G is f-choosable (or f-list-colorable) if it has a list coloring no matter how one assigns a list of f(v) colors to ...
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.
In graph theory, the Hadwiger conjecture states that if is loopless and has no minor then its chromatic number satisfies () <. It is known to be true for 1 ≤ t ≤ 6 {\displaystyle 1\leq t\leq 6} .
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, Brooks' theorem states a relationship between the maximum degree of a graph and its chromatic number. According to the theorem, in a connected graph in which every vertex has at most Δ neighbors, the vertices can be colored with only Δ colors, except for two cases, complete graphs and cycle graphs of odd length, which require ...
The total chromatic number of this graph is 6 since the degree of each vertex is 5 (5 adjacent edges + 1 vertex = 6). In graph theory , total coloring is a type of graph coloring on the vertices and edges of a graph.
The chromatic polynomial is a graph polynomial studied in algebraic graph theory, a branch of mathematics. It counts the number of graph colorings as a function of the number of colors and was originally defined by George David Birkhoff to study the four color problem .