4-Coloring H-Free Graphs When H Is Small

The k -Coloring problem is to test whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. If a graph G does not contain a graph H as an induced subgraph, then G is called H -free. For any fixed graph H on at most 6 vertices, it is known that 3-Coloring is polynomial-time solvable on H -free graphs whenever H is a linear forest and NP-complete otherwise. By solving the missing case P 2 +P 3 , we prove the same result for 4-Coloring provided that H is a fixed graph on at most 5 vertices.

[1]  Ingo Schiermeyer,et al.  3-Colorability in P for P6-free graphs , 2004, Discret. Appl. Math..

[2]  L. Lovász,et al.  Polynomial Algorithms for Perfect Graphs , 1984 .

[3]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[4]  Jian Song,et al.  Determining the chromatic number of triangle-free 2P3-free graphs in polynomial time , 2012, Theor. Comput. Sci..

[5]  Vadim V. Lozin,et al.  Deciding k-Colorability of P5-Free Graphs in Polynomial Time , 2007, Algorithmica.

[6]  Keith Edwards,et al.  The Complexity of Colouring Problems on Dense Graphs , 1986, Theor. Comput. Sci..

[7]  Zsolt Tuza,et al.  Graph colorings with local constraints - a survey , 1997, Discuss. Math. Graph Theory.

[8]  Myriam Preissmann,et al.  On the NP-completeness of the k-colorability problem for triangle-free graphs , 1996, Discret. Math..

[9]  Rajiv Raman,et al.  Colouring Vertices of Triangle-Free Graphs , 2010, WG.

[10]  Vadim V. Lozin,et al.  Coloring edges and vertices of graphs without short or long cycles , 2007, Contributions Discret. Math..

[11]  Jian Song,et al.  Coloring graphs without short cycles and long induced paths , 2011, Discret. Appl. Math..

[12]  Petr A. Golovach,et al.  List Coloring in the Absence of a Linear Forest , 2011, Algorithmica.

[13]  Jian Song,et al.  Updating the complexity status of coloring graphs without a fixed induced linear forest , 2012, Theor. Comput. Sci..

[14]  Zsolt Tuza,et al.  Complexity of Coloring Graphs without Forbidden Induced Subgraphs , 2001, WG.

[15]  Zvi Galil,et al.  NP Completeness of Finding the Chromatic Index of Regular Graphs , 1983, J. Algorithms.

[16]  Petr A. Golovach,et al.  Three complexity results on coloring Pk-free graphs , 2009, Eur. J. Comb..

[17]  Joe Sawada,et al.  A Certifying Algorithm for 3-Colorability of P5-Free Graphs , 2009, ISAAC.

[18]  Ingo Schiermeyer,et al.  Vertex Colouring and Forbidden Subgraphs – A Survey , 2004, Graphs Comb..

[19]  Vadim V. Lozin,et al.  Vertex 3-colorability of Claw-free Graphs , 2007, Algorithmic Oper. Res..

[20]  David Schindl,et al.  Some new hereditary classes where graph coloring remains NP-hard , 2005, Discret. Math..

[21]  Egon Balas,et al.  On graphs with polynomially solvable maximum-weight clique problem , 1989, Networks.

[22]  Petr A. Golovach,et al.  List Coloring in the Absence of a Linear Forest , 2011, WG.

[23]  Van Bang Le,et al.  On the complexity of 4-coloring graphs without long induced paths , 2007, Theor. Comput. Sci..

[24]  J. A. Bondy,et al.  Graph Theory , 2008, Graduate Texts in Mathematics.

[25]  Shuji Tsukiyama,et al.  A New Algorithm for Generating All the Maximal Independent Sets , 1977, SIAM J. Comput..

[26]  Gerhard J. Woeginger,et al.  The complexity of coloring graphs without long induced paths , 2001, Acta Cybern..