# Critical graph

In graph theory, a **critical graph** is a graph *G* in which every vertex or edge is a **critical element**, that is, if its deletion decreases the chromatic number of *G*. Such a decrease cannot be by more than 1.

*Not to be confused with the Critical path method, a concept in project management.*

## Variations

A ** k-critical graph** is a critical graph with chromatic number

*k*; a graph

*G*with chromatic number

*k*is

**if each of its vertices is a critical element. Critical graphs are the**

*k*-vertex-critical*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 of de Bruijn & Erdős 1951).- δ(
*G*) ≥*k*− 1, that is, every vertex is adjacent to at least*k*− 1 others. More strongly,*G*is (*k*− 1)-edge-connected. See Lovász (1992) - If
*G*is (*k*− 1)-regular, meaning every vertex is adjacent to exactly*k*− 1 others, then*G*is either*K*or an odd cycle . This is Brooks' theorem; see Brooks & Tutte (1941))._{k} - 2
*m*≥ (*k*− 1)*n*+*k*− 3 (Dirac 1957). - 2
*m*≥ (*k*− 1)*n*+ [(*k*− 3)/(*k*^{2}− 3)]*n*(Gallai 1963a). - 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 2*k*- 1 vertices (Gallai 1963b). 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 (Stehlík 2003).

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 Hajós (1961) 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. One open problem is to determine whether *K*_{k} is the only double-critical *k*-chromatic graph.[1]

## See also

## Sources

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*Proceedings of the Cambridge Philosophical Society*,**37**(02): 194–197, doi:10.1017/S030500410002168X - de Bruijn, N. G.; Erdős, P. (1951), "A colour problem for infinite graphs and a problem in the theory of relations",
*Nederl. Akad. Wetensch. Proc. Ser. A*,**54**: 371–373. (*Indag. Math.***13**.) - Dirac, G. A. (1957), "A theorem of R. L. Brooks and a conjecture of H. Hadwiger",
*Proceedings of the London Mathematical Society*,**7**(1): 161–195, doi:10.1112/plms/s3-7.1.161 - Erdős, Paul (1967), "Problem 2",
*In Theory of Graphs*, Proc. Colloq., Tihany, p. 361 - Gallai, T. (1963a), "Kritische Graphen I",
*Publ. Math. Inst. Hungar. Acad. Sci.*,**8**: 165–192 - Gallai, T. (1963b), "Kritische Graphen II",
*Publ. Math. Inst. Hungar. Acad. Sci.*,**8**: 373–395 - Hajós, G. (1961), "Über eine Konstruktion nicht n-färbbarer Graphen",
*Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe*,**10**: 116–117. - Jensen, T. R.; Toft, B. (1995),
*Graph coloring problems*, New York: Wiley-Interscience, ISBN 0-471-02865-7 - Stehlík, Matěj (2003), "Critical graphs with connected complements",
*Journal of Combinatorial Theory*, Series B,**89**(2): 189–194, doi:10.1016/S0095-8956(03)00069-8, MR 2017723. - Lovász, László (1992), "Solution to Exercise 9.21",
*Combinatorial Problems and Exercises (Second Edition)*, North-Holland - Stiebitz, Michael; Tuza, Zsolt; Voigt, Margit (6 August 2009), "On list critical graphs",
*Discrete Mathematics*, Elsevier,**309**(15): 4931–4941, doi:10.1016/j.disc.2008.05.021