In this paper,we survey known results on color-critical graphs in special graph classes.A graph is k-critical if its chromatic number is k but any proper subgraph of it has chromatic number less than k.For a family H ...In this paper,we survey known results on color-critical graphs in special graph classes.A graph is k-critical if its chromatic number is k but any proper subgraph of it has chromatic number less than k.For a family H of graphs,a graph is H-free if it does not contain H as an induced subgraph for every H∈H.A graph class is hereditary if it is H-free for some set H of graphs,and the graphs in H are called forbidden induced subgraphs for the class.We will focus on the characterization problem and the finiteness problem for hereditary graph classes that can be defined by one or two forbidden induced subgraphs.The characterization problem seeks a complete characterization of k-critical graphs in a given graph class and the finiteness problem asks if the number of k-critical graphs in a given class is finite.We shall survey results for both problems with an emphasis on how the results develop over the time and on the techniques used for proving results in the area.We also list important open problems and give some conjectures.展开更多
文摘In this paper,we survey known results on color-critical graphs in special graph classes.A graph is k-critical if its chromatic number is k but any proper subgraph of it has chromatic number less than k.For a family H of graphs,a graph is H-free if it does not contain H as an induced subgraph for every H∈H.A graph class is hereditary if it is H-free for some set H of graphs,and the graphs in H are called forbidden induced subgraphs for the class.We will focus on the characterization problem and the finiteness problem for hereditary graph classes that can be defined by one or two forbidden induced subgraphs.The characterization problem seeks a complete characterization of k-critical graphs in a given graph class and the finiteness problem asks if the number of k-critical graphs in a given class is finite.We shall survey results for both problems with an emphasis on how the results develop over the time and on the techniques used for proving results in the area.We also list important open problems and give some conjectures.
