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A certifying algorithm for 3-colorability of P5-free graphs
TL;DR: A certifying algorithm for the problem of deciding whether a P5-free graph is 3-colorable was proposed in this article, where it was shown that there are exactly six finite graphs that are not 3-colourable.
Abstract: We provide a certifying algorithm for the problem of deciding whether a P5- free graph is 3-colorable by showing there are exactly six finite graphs that are P5-free and not 3-colorable and minimal with respect to this property.
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TL;DR: In this article, the authors survey known results on the computational complexity of coloring and coloring for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)
eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
109 citations
TL;DR: Gyarfas proved the conjecture that for any tree T every T -free graph G with maximum clique size ω ( G ) is f T ( ω( G ) ) -colorable, for some function f T that depends only on T and υ ( G) .
23 citations
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TL;DR: In this article, it was shown that there are infinitely many 4-critical $H$-free graphs, if H is connected and not a subgraph of P_6.
Abstract: We prove that there are 24 4-critical $P_6$-free graphs, and give the complete list. We remark that, if $H$ is connected and not a subgraph of $P_6$, there are infinitely many 4-critical $H$-free graphs. Our result answers questions of Golovach et al. and Seymour.
9 citations
TL;DR: In this article, it was shown that a vertex or edge in a graph is critical if its deletion reduces the chromatic number of the graph by one, while an edge is critical only if its contraction reduces the number of vertices in the graph.
9 citations
TL;DR: It is proved that for any integers k, t>0 and a graph $H$ there are finitely many subgraph minimal graphs with no induced P_k and K_{t,t}$ that are not H-colorable.
8 citations
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TL;DR: In this article, a new method for accelerating matrix multiplication asymptotically is presented, based on the ideas of Volker Strassen, by using a basic trilinear form which is not a matrix product.
2,454 citations
TL;DR: The strong perfect graph conjecture as discussed by the authors states that a graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced sub graph of G is an odd cycle of length at least five or the complement of one.
Abstract: A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one. The ?strong perfect graph conjecture? (Berge, 1961) asserts that a graph is perfect if and only if it is Berge. A stronger conjecture was made recently by Conforti, Cornu?ejols and Vuiskovi?c ? that every Berge graph either falls into one of a few basic classes, or admits one of a few kinds of separation (designed so that a minimum counterexample to Berge?s conjecture cannot have either of these properties). In this paper we prove both of these conjectures.
1,161 citations
14 Jun 2001
TL;DR: A complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete is given.
Abstract: We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem for two forbidden subgraphs.
218 citations
TL;DR: In this paper, it was shown that for every fixed integer k, there exists a polynomial-time algorithm for determining whether a P5-free graph admits a k-coloring, and finding one, if it does.
Abstract: The problem of computing the chromatic number of a P5-free graph (a graph which contains no path on 5 vertices as an induced subgraph) is known to be NP-hard. However, we show that for every fixed integer k, there exists a polynomial-time algorithm determining whether or not a P5-free graph admits a k-coloring, and finding one, if it does.
160 citations
TL;DR: This work gives linear-time certifying algorithms for recognition of interval graphs and permutation graphs and shows that their certificates of non-membership can be authenticated in O(|V|) time.
Abstract: A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V|) time.
139 citations