Vertex-Coloring with Star-Defects

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors.

[1]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[2]  Douglas R. Woodall,et al.  Defective colorings of graphs in surfaces: Partitions into subgraphs of bounded valency , 1986, J. Graph Theory.

[3]  Mehdi Behzad,et al.  Graphs and Digraphs , 1981, The Mathematical Gazette.

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

[5]  Paul Dorbec,et al.  Vertex Partitions of Graphs into Cographs and Stars , 2014, J. Graph Theory.

[6]  David P. Dailey.  Uniqueness of colorability and colorability of planar 4-regular graphs are NP-complete , 1980, Discret. Math..

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

[8]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

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

[10]  Wayne Goddard,et al.  Defective coloring revisited , 1997, J. Graph Theory.

[11]  A. Hajnal,et al.  On decomposition of graphs , 1967 .

[12]  Saumyendra Sengupta,et al.  Graphs and Digraphs , 1994 .

[13]  Krzysztof Onak,et al.  An Efficient Partitioning Oracle for Bounded-Treewidth Graphs , 2011, APPROX-RANDOM.

[14]  Dan Archdeacon,et al.  A note on defective colorings of graphs in surfaces , 1987, J. Graph Theory.

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

[16]  G. Chartrand,et al.  A generalization of the chromatic number , 1968, Mathematical Proceedings of the Cambridge Philosophical Society.

[17]  Lukasz Kowalik,et al.  Fast 3-coloring Triangle-Free Planar Graphs , 2004, Algorithmica.

[18]  David R. Wood,et al.  A Note on Tree-Partition-Width , 2006 .

[19]  Siam J. CoMPtrr,et al.  FINDING A MAXIMUM CUT OF A PLANAR GRAPH IN POLYNOMIAL TIME * , 2022 .

[20]  Fan Chung,et al.  On the Decomposition of Graphs , 1981 .

[21]  Pieter Maritz,et al.  Francis Guthrie: A Colourful Life , 2012, The Mathematical Intelligencer.

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