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.
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 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 Kk or an odd cycle . This is Brooks' theorem; see Brooks & Tutte (1941)).
- 2 m ≥ (k − 1) n + k − 3 (Dirac 1957).
- 2 m ≥ (k − 1) n + [(k − 3)/(k2 − 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 2k - 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 Kk 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 Kk is the only double-critical k-chromatic graph.
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- Erdős, Paul (1967), "Problem 2", In Theory of Graphs, Proc. Colloq., Tihany, p. 361
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- 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