In graph theory, a critical graph is an undirected graph all of whose proper subgraphs have smaller chromatic number. In such a graph, every vertex or edge is a critical element, in the sense that its deletion would decrease the number of colors needed in a graph coloring of the given graph. Each time a single edge or vertex (along with its incident edges) is removed from a critical graph, the decrease in the number of colors needed to color that graph cannot be by more than one.

Variations

A -critical graph is -vertex-critical if each of its vertices is a critical element. Critical graphs are the minimal members in terms of chromatic number, which is a very important measure in graph theory.

edges:

• has only one component.

• is finite (this is the De Bruijn–Erdős theorem).

• The minimum degree-edge-connected.

• is a regular graph is either the complete graph vertices, or an odd-length cycle graph. This is Brooks' theorem.

• .

• .

• vertices. is the only vertex of its color and every other color class has at least two vertices.

is vertex-critical if and only if is a singleton color class.

As -critical graph may be formed from a complete graph by combining the Hajós construction colors in any proper coloring.

A double-critical graph is a connected graph in which the deletion of any pair of adjacent vertices decreases the chromatic number by two. It is an open problem-chromatic graph.

See also

• Factor-critical graph

References

Further reading