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. 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 ${\displaystyle k}$-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).
• 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.
• 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.
• ${\displaystyle 2m\geq (k-1)n+k-3}$.
• ${\displaystyle 2m\geq (k-1)n+(k-3)/(k^{2}-3)n}$.
• 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. 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.

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.

Dirac's critical-graph conjecture asks whether, for every integer k ≥ 4, there is a k-vertex-critical graph with no critical edge. Tommy R. Jensen established this for every k ≥ 5 in 2002; a 2026 working paper by Alex Chan completes the conjecture by offering a construction for the remaining case k = 4.

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.

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 (as claimed by the Erdös-Lovász double critical conjecture).

See also

• Factor-critical graph

References

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Further reading