Acyclic coloring

The acyclic chromatic number of McGee graph is 3.

In graph theory, an acyclic coloring is a (proper) vertex coloring in which every 2-chromatic subgraph is acyclic. The acyclic chromatic number A(G) of a graph G is the least number of colors needed in any acyclic coloring of G.

Acyclic coloring is often associated with graphs embedded on non-plane surfaces.

Upper Bounds

A(G) ≤ 2 if and only if G is acyclic.

Bounds on A(G) in terms of the maximum degree Δ(G) of G include the following:

• A(G) ≤ 4 if Δ(G) = 3. (Grünbaum 1973)
• A(G) ≤ 5 if Δ(G) = 4. (Burstein 1979)
• A(G) ≤ 7 if Δ(G) = 5. (Kostochka & Stocker 2011)
• A(G) ≤ 12 if Δ(G) = 6. (Yadav & Satish 2009)

A milestone in the study of acyclic coloring is the following affirmative answer to a conjecture of Grünbaum:

Theorem. (Borodin 1979)

A(G) ≤ 5 if G is planar graph.

Grünbaum (1973) introduced acyclic coloring and acyclic chromatic number, and conjectured the result in the above theorem. Borodin's proof involved several years of painstaking inspection of 450 reducible configurations. One consequence of this theorem is that every planar graph can be decomposed into an independent set and two induced forests. (Stein 1970, 1971)

Algorithms and Complexity

It is NP-complete to determine whether A(G) ≤ 3. (Kostochka 1978)

Coleman & Cai (1986) showed that the decision variant of the problem is NP-complete even when G is a bipartite graph.

Gebremedhin et al. (2008) demonstrated that every proper vertex coloring of a chordal graph is also an acyclic coloring. Since chordal graphs can be optimally colored in O(n+m) time, the same is also true for acyclic coloring on that class of graphs.

A linear-time algorithm to acyclically color a graph of maximum degree ≤ 3 using 4 colors or fewer was given by Skulrattanakulchai (2004).

See also

• Star coloring

References

• Borodin, O. V. (1979), "On acyclic colorings of planar graphs", Discrete Mathematics, 25: 211–236, doi:10.1016/0012-365X(79)90077-3 .
• Burstein, M. I. (1979), "Every 4-valent graph has an acyclic 5-coloring (in Russian)", Soobšč. Akad. Nauk Gruzin. SSR, 93: 21–24. 
• Grünbaum, B. (1973), "Acyclic colorings of planar graphs", Israel J. Math., 14: 390–408, doi:10.1007/BF02764716. 
• Coleman, Thomas F.; Cai, Jin-Yi (1986), "The Cyclic Coloring Problem and Estimation of Sparse Hessian Matrices", SIAM. J. on Algebraic and Discrete Methods, 7 (2): 221–235, doi:10.1137/0607026 .
• Fertin, Guillaume; Raspaud, André (2008), "Acyclic coloring of graphs of maximum degree five: Nine colors are enough", Information Processing Letters, 105 (2): 65–72, doi:10.1016/j.ipl.2007.08.022 .
• Gebremedhin, Assefaw H.; Tarafdar, Arijit; Pothen, Alex; Walther, Andrea (2008), "Efficient Computation of Sparse Hessians Using Coloring and Automatic Differentiation", Informs Journal on Computing, 21: 209, doi:10.1287/ijoc.1080.0286 .
• Jensen, Tommy R.; Toft, Bjarne (1995), Graph Coloring Problems, New York: Wiley-Interscience, ISBN 0-471-02865-7 .
• Kostochka, A. V. (1978), Upper bounds of chromatic functions of graphs, Doctoral thesis (in Russian), Novosibirsk .
• Kostochka, Alexandr V.; Stocker, Christopher (2011), "Graphs with maximum degree 5 are acyclically 7-colorable", Ars Mathematica Contemporanea, 4 (1): 153–164, MR 2785823 .
• Skulrattanakulchai, San (2004), "Acyclic colorings of subcubic graphs", Information Processing Letters, 92 (4): 161–167, doi:10.1016/j.ipl.2004.08.002 .
• Stein, S. K. (1970), "B-sets and coloring problems", Bull. Amer. Math. Soc., 76: 805–806, doi:10.1090/S0002-9904-1970-12559-9. 
• Stein, S. K. (1971), "B-sets and planar maps", Pacific J. Math., 37 (1): 217–224, doi:10.2140/pjm.1971.37.217. 
• Yadav, Kishore; Satish, Venkaiah; Yadav, Kishore; Kothapalli, Kishore (2009), "Acyclic coloring of graphs of maximum degree six: Twelve colors are enough", Electronic Notes in Discrete Mathematics, 35: 177–182, doi:10.1016/j.endm.2009.11.030 .

External links

• Star colorings and acyclic colorings (1973), present at the Research Experiences for Graduate Students (REGS) at the University of Illinois, 2008.
• Acyclic Coloring of Graphs of Maximum Degree ∆, talk slides presented by G. Fertin and A. Raspaud at EUROCOMB 05, Berlin, 2005.