Coloring outerplanar graphs and planar 3-trees with small monochromatic components

In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.

[1]  Nobuji Saito,et al.  Linear-time computability of combinatorial problems on series-parallel graphs , 1982, JACM.

[2]  David Eppstein,et al.  3-Coloring in Time O(1.3289^n) , 2000, J. Algorithms.

[3]  P. Pardalos,et al.  The Graph Coloring Problem: A Bibliographic Survey , 1998 .

[4]  Ping Wang,et al.  A note on total colorings of planar graphs without 4-cycles , 2004, Discuss. Math. Graph Theory.

[5]  Robert Berke,et al.  Colorings and transversals of graphs , 2008 .

[6]  Donald MacKenzie,et al.  Mechanizing Proof: Computing, Risk, and Trust , 2001 .

[7]  Gábor Tardos,et al.  Bounded size components--partitions and transversals , 2003, J. Comb. Theory, Ser. B.

[8]  N Linial,et al.  Low diameter graph decompositions , 1993, Comb..

[9]  Gwenaël Joret,et al.  Colouring Planar Graphs With Three Colours and No Large Monochromatic Components , 2013, Combinatorics, Probability and Computing.

[10]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[11]  Richard Cole,et al.  New Linear-Time Algorithms for Edge-Coloring Planar Graphs , 2007, Algorithmica.

[12]  L. Lovász.  Three short proofs in graph theory , 1975 .

[13]  Eugene L. Lawler,et al.  A Note on the Complexity of the Chromatic Number Problem , 1976, Inf. Process. Lett..

[14]  Noga Alon,et al.  Partitioning into graphs with only small components , 2003, J. Comb. Theory, Ser. B.

[15]  Jirí Matousek,et al.  Graph coloring with no large monochromatic components , 2007, Electron. Notes Discret. Math..

[16]  Sal Restivo.  Mechanizing Proof: Computing, Risk, and Trust, by D. MacKenzie. Cambridge, MA: MIT Press, 2001. xi + 427 pp. $50.00. ISBN 0-262-13393-8. , 2004 .

[17]  Bogdan Oporowski,et al.  On tree-partitions of graphs , 1996, Discret. Math..

[18]  Fedor V. Fomin,et al.  Improved Exact Algorithms for Counting 3- and 4-Colorings , 2007, COCOON.

[19]  L. Cowen,et al.  Defective coloring revisited , 1997 .

[20]  Bruce A. Reed,et al.  On Total Colorings of Graphs , 1993, J. Comb. Theory, Ser. B.

[21]  Andrzej Proskurowski,et al.  Efficient vertex- and edge-coloring of outerplanar graphs , 1986 .

[22]  Herbert S. Wilf,et al.  Backtrack: An O(1) Expected Time Algorithm for the Graph Coloring Problem , 1984, Inf. Process. Lett..

[23]  D. Gale.  The Game of Hex and the Brouwer Fixed-Point Theorem , 1979 .

[24]  Tibor Szabó,et al.  Relaxed two-coloring of cubic graphs , 2007, J. Comb. Theory, Ser. B.

[25]  Georges Gonthier,et al.  Formal Proof—The Four- Color Theorem , 2008 .

[26]  Jirí Matousek,et al.  Large Monochromatic Components in Two-colored Grids , 2007, Electron. Notes Discret. Math..

[27]  Jianfeng Hou,et al.  On total colorings of 1-planar graphs , 2013, J. Comb. Optim..

[28]  Rajeev Motwani,et al.  Storage management for evolving databases , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[29]  Andreas Björklund,et al.  Set Partitioning via Inclusion-Exclusion , 2009, SIAM J. Comput..

[30]  Marek Chrobak,et al.  Fast Algorithms for Edge-Coloring Planar Graphs , 1989, J. Algorithms.