Critical graph



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. The decrease in the number of colors cannot be by more than one.

Variations
A $$k$$-critical graph is a critical graph with chromatic number $$k$$. A graph $$G$$ with chromatic number $$k$$ is $$k$$-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.

Some properties of a $$k$$-critical graph $$G$$ with $$n$$ vertices and $$m$$ edges:
 * $$G$$ has only one component.
 * $$G$$ is finite (this is the de Bruijn–Erdős theorem).
 * The minimum degree $$\delta(G)$$ obeys the inequality $$\delta(G)\ge k-1$$. That is, every vertex is adjacent to at least $$k-1$$ others. More strongly, $$G$$ is $$(k-1)$$-edge-connected.
 * If $$G$$ is a regular graph with degree $$k-1$$, meaning every vertex is adjacent to exactly $$k-1$$ others, then $$G$$ is either the complete graph $$K_k$$ with $$n=k$$ vertices, or an odd-length cycle graph. This is Brooks' theorem.
 * $$2m\ge(k-1)n+k-3$$.
 * $$2m\ge (k-1)n+\lfloor(k-3)/(k^2-3)\rfloor n$$.
 * Either $$G$$ may be decomposed into two smaller critical graphs, with an edge between every pair of vertices that includes one vertex from each of the two subgraphs, or $$G$$ has at least $$2k-1$$ vertices. More strongly, either $$G$$ has a decomposition of this type, or for every vertex $$v$$ of $$G$$ there is a $$k$$-coloring in which $$v$$ is the only vertex of its color and every other color class has at least two vertices.

Graph $$G$$ is vertex-critical if and only if for every vertex $$v$$, there is an optimal proper coloring in which $$v$$ is a singleton color class.

As showed, every $$k$$-critical graph may be formed from a complete graph $$K_k$$ by combining the Hajós construction with an operation that identifies two non-adjacent vertices. The graphs formed in this way always require $$k$$ 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 to determine whether $$K_k$$ is the only double-critical $$k$$-chromatic graph.