Critical graph

A ${\displaystyle k}$-critical graph is a critical graph with chromatic number ${\displaystyle k}$. A graph ${\displaystyle G}$ with chromatic number ${\displaystyle k}$ is ${\displaystyle k}$-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 ${\displaystyle k}$-critical graph ${\displaystyle G}$ with ${\displaystyle n}$ vertices and ${\displaystyle m}$ edges:

• ${\displaystyle G}$ has only one component.
• ${\displaystyle G}$ is finite (this is the de Bruijn–Erdős theorem).[1]
• The minimum degree ${\displaystyle \delta (G)}$ obeys the inequality ${\displaystyle \delta (G)\geq k-1}$. That is, every vertex is adjacent to at least ${\displaystyle k-1}$ others. More strongly, ${\displaystyle G}$ is ${\displaystyle (k-1)}$-edge-connected.[2]
• If ${\displaystyle G}$ is a regular graph with degree ${\displaystyle k-1}$, meaning every vertex is adjacent to exactly ${\displaystyle k-1}$ others, then ${\displaystyle G}$ is either the complete graph ${\displaystyle K_{k}}$ with ${\displaystyle n=k}$ vertices, or an odd-length cycle graph. This is Brooks' theorem.[3]
• ${\displaystyle 2m\geq (k-1)n+k-3}$.[4]
• ${\displaystyle 2m\geq (k-1)n+\lfloor (k-3)/(k^{2}-3)\rfloor n}$.[5]
• Either ${\displaystyle G}$ 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 ${\displaystyle G}$ has at least ${\displaystyle 2k-1}$ vertices.[6] More strongly, either ${\displaystyle G}$ has a decomposition of this type, or for every vertex ${\displaystyle v}$ of ${\displaystyle G}$ there is a ${\displaystyle k}$-coloring in which ${\displaystyle v}$ is the only vertex of its color and every other color class has at least two vertices.[7]

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

As Hajós (1961) showed, every ${\displaystyle k}$-critical graph may be formed from a complete graph ${\displaystyle K_{k}}$ by combining the Hajós construction with an operation that identifies two non-adjacent vertices. The graphs formed in this way always require ${\displaystyle k}$ colors in any proper coloring.[8]

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 ${\displaystyle K_{k}}$ is the only double-critical ${\displaystyle k}$-chromatic graph.[9]

• Factor-critical graph