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Star Coloring of Graphs with Girth at Least Five

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Abstract

For two disjoint vertex subsets X, Y of a graph G, we denote \(X \leftarrow Y\) if every vertex of Y has at most one non-neighbour in X. A k-clique star partition of a graph G is \(V(G)=Q_1\cup Q_2\cup \ldots \cup Q_k\) such that (i) \(Q_{i}\) is a clique in G for all \(1\le i \le k\) and (ii) \(Q_i\leftarrow Q_j\) for all \(1 \le i < j\le k\). We prove that (a) every \(\{3K_1, 2K_2 \}\)-free graph admits a \(4\omega (G)\)-clique star partition and (b) if G is a graph with girth at least five, then its star chromatic number \(\chi _s (G)\) satisfies \(\chi _s (G) \le 4 \alpha (G)\).

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References

  1. Albertson, M.O., Chappell, G.G., Kierstead, H.A., Kündgen, A., Ramamurthi, R.: Coloring with no 2-colored \(P_4\)’s. Electron. J. Comb. 11, R26 (2004)

    MATH  Google Scholar 

  2. Aravind, N.R., Subramanian, C.R.: Bounds on vertex colorings with restrictions on the union of color classes. J. Graph Theory 66, 213–234 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blázsik, Z., Hujter, M., Pluhár, A., Tuza, Z.: Graphs with no induced \(C_{4}\) and \(2K_{2}\). Discr. Math. 115, 51–55 (1993)

    Article  MATH  Google Scholar 

  4. Borodin, O.V., Kostochka, A.V., Woodall, D.R.: Acyclic coloring of planar graphs with large girth. J. Lond. Math. Soc. 60, 344–352 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, M., Raspaud, A., Wang, W.: 6-Star-coloring of subcubic graphs. J. Graph Theory 72, 128–145 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fertin, G., Raspaud, A., Reed, B.: Star coloring of graphs. J. Graph Theory 47, 163–182 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Földes, S., Hammer, P.L.: Split graphs. Congr. Numer. 19, 311–315 (1977)

    MathSciNet  MATH  Google Scholar 

  8. Gebremedhin, A.H., Tarafdar, A., Manne, F., Pothen, A.: New acyclic and star coloring algorithms with applications to computing Hessians. SIAM J. Sci. Comput. 29, 1042–1072 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Grünbaum, B.: Acyclic colorings of planar graphs. Isr. J. Math. 14, 390–408 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kawarabayashi, K., Mohar, B.: Star coloring and acyclic coloring of locally planar graphs. SIAM J. Discr. Math. 24, 56–71 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kierstead, H.A., Kündgen, A., Timmons, C.: Star coloring bipartite planar graphs. J. Graph Theory 60, 1–10 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kündgen, A., Timmons, C.: Star coloring planar graphs from small lists. J. Graph Theory 63, 324–337 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Lyons, A.: Acyclic and star colorings of cographs. Discr. Appl. Math. 159, 1842–1850 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nešetřil, J., de Mendez, P.O.: Colorings and homomorphisms of minor closed classes. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds.) Discrete and Computational Geometry: The GoodmanPollack Festschrift, pp. 651–664. Springer, New York (2003)

    Google Scholar 

  15. Timmons, C.: Star coloring high girth planar graphs. Electron. J. Comb. 15, R124 (2008)

    MathSciNet  MATH  Google Scholar 

  16. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Upper Saddle River (2000)

    Google Scholar 

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Acknowledgments

The authors wish to thank the anonymous referees whose suggestions improved the presentation of this paper.

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  1. Indian Institute of Information Technology, Design and Manufacturing (IIITD&M), Kancheepuram, Chennai, 600127, India

    M. A. Shalu & T. P. Sandhya

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  1. M. A. Shalu
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  2. T. P. Sandhya
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Correspondence to M. A. Shalu.

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Shalu, M.A., Sandhya, T.P. Star Coloring of Graphs with Girth at Least Five. Graphs and Combinatorics 32, 2121–2134 (2016). https://doi.org/10.1007/s00373-016-1702-2

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  • DOI: https://doi.org/10.1007/s00373-016-1702-2

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