ESA 2006: Algorithms – ESA 2006 pp 124-135

# Deciding Relaxed Two-Colorability—A Hardness Jump

• Robert Berke
• Tibor Szabó
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4168)

## Abstract

A coloring is proper if each color class induces connected components of order one (where the order of a graph is its number of vertices). Here we study relaxations of proper two-colorings, such that the order of the induced monochromatic components in one (or both) of the color classes is bounded by a constant. In a (C 1 , C 2 )-relaxed coloring of a graph G every monochromatic component induced by vertices of the first (second) color is of order at most C 1 (C 2, resp.). We are mostly concerned with (1,C)-relaxed colorings, in other words when/how is it possible to break up a graph into small components with the removal of an independent set.

We prove that every graph of maximum degree at most three can be (1,22)-relaxed colored and we give a quasilinear algorithm which constructs such a coloring. We also show that a similar statement cannot be true for graphs of maximum degree at most 4 in a very strong sense: we construct 4-regular graphs such that the removal of any independent set leaves a connected component whose order is linear in the number of vertices.

Furthermore we investigate the complexity of the decision problem (Δ, C)-AsymRelCol: Given a graph of maximum degree at most Δ, is there a (1,C)-relaxed coloring of G? We find a remarkable hardness jump in the behavior of this problem. We note that there is not even an obvious monotonicity in the hardness of the problem as C grows, i.e. the hardness for component order C+1 does not imply directly the hardness for C. In fact for C=1 the problem is obviously polynomial-time decidable, while it is shown that it is NP-hard for C=2 and Δ≥3.

For arbitrary Δ≥2 we still establish the monotonicity of hardness of (Δ, C)-AsymRelCol on the interval 2≤C ≤∞ in the following strong sense. There exists a critical component order f(Δ)∈ℕ∪{∞} such that the problem of deciding (1,C)-relaxed colorability of graphs of maximum degree at most Δ is NP-complete for every 2≤C< f(Δ), while deciding (1,f(Δ))-colorability is trivial: every graph of maximum degree Δ is (1,f(Δ))-colorable. For Δ=3 the existence of this threshold is shown despite the fact that we do not know its precise value, only 6≤f(3)≤22. For any Δ≥4, (Δ, C)-AsymRelCol is NP-complete for arbitrary C≥2, so f(Δ)=∞.

We also study the symmetric version of the relaxed coloring problem, and make the first steps towards establishing a similar hardness jump.

## Keywords

Perfect Match Maximum Degree Chromatic Number Conjunctive Normal Form Full Version
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Alon, N., Ding, G., Oporowski, B., Vertigan, D.: Partitioning into graphs with only small components. Journal of Combinatorial Theory (Series B) 87, 231–243 (2003)
2. 2.
Akiyama, J., Chvátal, V.: A short proof of the linear arboricity for cubic graphs. Bull. Liber. Arts Sci. NMS 2 (1981)Google Scholar
3. 3.
Akiyama, J., Exoo, G., Harary, F.: Coverings and packings in graphs II. Cyclic and acyclic invariants. Math. Slovaca 30, 405–417 (1980)
4. 4.
Berke, R., Szabó, T.: Deciding Relaxed Two-Colorability — A Hardness Jump, full version, available at: http://www.inf.ethz.ch/personal/berker/publications/relcom-full.pdf
5. 5.
Berke, R., Szabó, T.: Relaxed Coloring of Cubic Graphs. In: Proceedings EuroComb., (extended abstract), pp. 341–344 (submitted, 2005)Google Scholar
6. 6.
Biedl, T.C., Bose, P., Demaine, E.D., Lubiw, A.: Efficient Algorithms for Petersen’s Matching Theorem. Journal of Algorithms 38, 110–134 (2001)
7. 7.
Edwards, K., Farr, G.: Fragmentability of graphs. J. Comb. Th. Series B 82(1), 30–37 (2001)
8. 8.
Havet, F., Sereni, J.-S.: Improper choosability of graphs and maximum average degree. J. Graph Theory (to appear, 2005)Google Scholar
9. 9.
Havet, F., Kang, R., Sereni, J.-S.: Improper colouring of unit disk graphs. Proceedings ICGT 22, 123–128 (2005)Google Scholar
10. 10.
Haxell, P., Szabó, T., Tardos, G.: Bounded size components — partitions and transversals. Journal of Combinatorial Theory (Series B) 88(2), 281–297 (2003)
11. 11.
Haxell, P., Pikhurko, O., Thomason, A.: Maximum Acyclic and Fragmented Sets in Regular Graphs (submitted)Google Scholar
12. 12.
Hoory, S., Szeider, S.: Families of unsatisfiable k-CNF formulas with few occurrences per variable. arXiv.org e-Print archive, math. CO/0411167 (2004)Google Scholar
13. 13.
Hoory, S., Szeider, S.: Computing Unsatisfiable k-SAT Instances with Few Occurrences per Variable. Theoretical Computer Science 337(1-3), 347–359 (2005)
14. 14.
Kratochvíl, J., Savický, P., Tuza, Z.: One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM J. Comput. 22(1), 203–210 (1993)
15. 15.
Linial, N., Saks, M.: Low diameter graph decompositions. Combinatorica 13(4), 441–454 (1993)
16. 16.
Savický, P., Sgall, J.: DNF tautologies with a limited number of of occurrences of every variable. Theoret. Comput. Sci. 238(1-2), 495–498 (2000)
17. 17.
Škrekovski, R.: List improper colorings of planar graphs. Comb. Prob. Comp. 8, 293–299 (1999)
18. 18.
Thomassen, C.: Two-colouring the edges of a cubic graph such that each monochromatic component is a path of length at most 5. J. Comb. Th. Series B 75, 100–109 (1999)
19. 19.
Tovey, C.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984)