Three-Coloring and List Three-Coloring of Graphs Without Induced Paths on Seven Vertices

In this paper we present a polynomial time algorithm that determines if an input graph containing no induced seven-vertex path is 3-colorable. This affirmatively answers a question posed by Randerath, Schiermeyer and Tewes in 2002. Our algorithm also solves the list-coloring version of the 3-coloring problem, where every vertex is assigned a list of colors that is a subset of {1,2,3}, and gives an explicit coloring if one exists.

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

[2]  Maria Chudnovsky,et al.  4‐Coloring P6‐Free Graphs with No Induced 5‐Cycles , 2014, J. Graph Theory.

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

[4]  Shenwei Huang,et al.  Complexity of coloring graphs without paths and cycles , 2013, Discret. Appl. Math..

[5]  Shenwei Huang.  Improved Complexity Results on k-Coloring P t -Free Graphs , 2013, MFCS.

[6]  Daniël Paulusma,et al.  Narrowing the Complexity Gap for Colouring (Cs, Pt)-Free Graphs , 2015, Comput. J..

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

[8]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[9]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..

[10]  Ingo Schiermeyer,et al.  Three-colourability and forbidden subgraphs. II: polynomial algorithms , 2002, Discret. Math..

[11]  Eglantine Camby,et al.  A New Characterization of $$P_k$$Pk-Free Graphs , 2015, Algorithmica.

[12]  Jian Song,et al.  A Survey on the Computational Complexity of Coloring Graphs with Forbidden Subgraphs , 2014, J. Graph Theory.

[13]  Larry Stockmeyer,et al.  Planar 3-colorability is polynomial complete , 1973, SIGA.

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

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

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

[17]  Jian Song,et al.  Closing Complexity Gaps for Coloring Problems on H-Free Graphs , 2012, ISAAC.

[18]  Shenwei Huang,et al.  Improved complexity results on k-coloring Pt-free graphs , 2013, Eur. J. Comb..

[19]  Eglantine Camby,et al.  A New Characterization of Pk-free Graphs , 2014, WG.

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

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

[22]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[23]  Eglantine Camby,et al.  A New Characterization of P k -free Graphs. , 2014, WG 2014.

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

[25]  Daniël Paulusma,et al.  Narrowing the Complexity Gap for Colouring (C s , P t )-Free Graphs , 2014, AAIM.