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Ji rbreve

Bio: Ji rbreve is an academic researcher. The author has contributed to research in topics: 1-planar graph & Edge coloring. The author has an hindex of 1, co-authored 1 publications receiving 68 citations.

Papers
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Journal Article
TL;DR: NP-completeness of deciding whether a Ps-free graph is 5-colorable is proved and a polynomial time algorithm is given for deciding whether an Pi2-free graphs is 4- colorable.
Abstract: We discuss the computational complexity of determining the chromatic number of graphs without long induced paths. We prove NP-completeness of deciding whether a Ps-free graph is 5-colorable and of deciding whether a Pi2-free graph is 4-colorable. Moreover, we give a polynomial time algorithm for deciding whether a Ps-free graph is 3-colorable.

69 citations


Cited by
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Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: This work surveys results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions in order to obtain useful results from a graph coloring formulation of his problem.
Abstract: There is a great variety of colouring concepts and results in the literature. Here our focus is to survey results on vertex colourings of graphs defined in terms of forbidden induced subgraph conditions. Thus, one who wishes to obtain useful results from a graph coloring formulation of his problem must do more than just show that the problem is equivalent to the general problem of coloring a graph. If there is to be any hope, one must also obtain information about the structure of the graphs that need to be colored (D.S. Johnson [66]).

150 citations

Journal ArticleDOI
TL;DR: In this article, the authors survey known results on the computational complexity of k-coloring and k-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 coloring, or where lists of permissible colors are given for each vertex.
Abstract: For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING 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 coloring, or where lists of permissible colors are given for each vertex.

128 citations

Posted Content
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

Journal ArticleDOI
TL;DR: It is shown that both vertex and edge colorability problems remain difficult even for graphs without short cycles, i.e., without cycles of length at most g for any particular value of g, and that both problems are shown to be solvable in poynomial time.
Abstract: Vertex and edge colorability are two graph problems that are NP-hard in general. We show that both problems remain difficult even for graphs without short cycles, i.e., without cycles of length at most g for any particular value of g. On the contrary, for graphs without long cycles, both problems are shown to be solvable in poynomial time.

104 citations