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The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups

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Abstract

A k-L(2,1)-labelling of a graph G is a mapping f:V(G)→{0,1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(G) and f(u)≠f(v) if u,v are distance two apart. The smallest positive integer k such that G admits a k-L(2,1)-labelling is called the λ-number of G. In this paper we study this quantity for cubic Cayley graphs (other than the prism graphs) on dihedral groups, which are called brick product graphs or honeycomb toroidal graphs. We prove that the λ-number of such a graph is between 5 and 7, and moreover we give a characterisation of such graphs with λ-number 5.

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References

  • Alspach B, Chen CC, Mcavanney K (1996) On a class of Hamiltonian laceable 3-regular graphs. Discrete Math 151:19–38

    Article  MathSciNet  MATH  Google Scholar 

  • Alspach B, Dean M (2009) Honeycomb toroidal graphs are Cayley graphs. Inf Process Lett 109:705–708

    Article  MathSciNet  MATH  Google Scholar 

  • Alspach B, Zhang CQ (1989) Hamilton cycles in cubic graphs on dihedral groups. Ars Comb 28:101–108

    MathSciNet  MATH  Google Scholar 

  • Bahls P (2011) Channel assignment on Cayley graphs. J Graph Theory 67(3):169–177

    Article  MathSciNet  MATH  Google Scholar 

  • Bodlaender HL, Kloks T, Tan RB, van Leeuwen J (2004) Approximations for λ-coloring of graphs. Comput J 47:193–204

    Article  MATH  Google Scholar 

  • Calamoneri T, Petreschi R (2004) L(h,1)-labeling subclasses of planar graphs. J Parallel Distrib Comput 64(3):414–426.

    Article  MATH  Google Scholar 

  • Calamoneri T (2011) The L(h,k)-labelling problem: an updated survey and annotated bibliography. Comput J 54:1344–1371

    Article  Google Scholar 

  • Calamoneri T, Caminiti S, Petreschi R (2008) A general approach to L(h,k)-label interconnection networks. J Comput Sci Technol 23(4):652–659

    Article  MathSciNet  Google Scholar 

  • Chang GJ, Kuo D (1996) The L(2,1)-labelling problem on graphs. SIAM J Discrete Math 9:309–316

    Article  MathSciNet  MATH  Google Scholar 

  • Chang GJ, Lu CH, Zhou S (2009) Distance-two labellings of Hamming graphs. Discrete Appl Math 157:1896–1904

    Article  MathSciNet  MATH  Google Scholar 

  • Chang GJ, Lu CH, Zhou S (2007) No-hole 2-distant colorings for Cayley graphs on finitely generated Abelian groups. Discrete Math 307:1808–1817

    Article  MathSciNet  MATH  Google Scholar 

  • Cho H, Hsu L (2003) Generalized honeycomb torus. Inf Process Lett 86:185–190

    Article  MathSciNet  MATH  Google Scholar 

  • Coxeter HSM (1950) Self dual configurations and regular graphs. Bull Am Math Soc 56:413–455

    Article  MathSciNet  MATH  Google Scholar 

  • Georges JP, Mauro DW (2002) On generalized Petersen graphs labelled with a condition at distance two. Discrete Math 259:311–318

    Article  MathSciNet  MATH  Google Scholar 

  • Goncalves D (2008) On the L(p,1)-labelling of graphs. Discrete Math 308:1405–1414

    Article  MathSciNet  MATH  Google Scholar 

  • Griggs JR, Yeh RK (1992) Labelling graphs with a condition at distance 2. SIAM J Discrete Math 5:586–595

    Article  MathSciNet  MATH  Google Scholar 

  • Hale WK (1980) Frequency assignment: Theory and applications. Proc IEEE 68:1497–1514

    Article  Google Scholar 

  • Havet F, Reed B, Sereni J-S (2012) Griggs and Yeh’s conjecture and L(p,1)-labelings. SIAM J Discrete Math 26:145–168

    Article  MathSciNet  MATH  Google Scholar 

  • Jha PK, Narayanan A, Sood P, Sundaram K, Sunder V (2000) On L(2;1)-labeling of the Cartesian product of a cycle and a path. Ars Comb 55:81–89

    MathSciNet  MATH  Google Scholar 

  • Kang J-H (2008) L(2,1)-labelling of Hamiltonian graphs with maximum degree 3. SIAM J Discrete Math 22:213–230

    Article  MathSciNet  MATH  Google Scholar 

  • Král’ D, Škrekovski R (2003) A theorem about the channel assignment problem. SIAM J Discrete Math 16:426–437

    Article  MathSciNet  MATH  Google Scholar 

  • Kuo D, Yan J-H (2004) On L(2,1)-labellings of Cartesian products of paths and cycles. Discrete Math 283:137–144

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Mak-Hau V, Zhou S (2012) Distance-two labellings for chordal rings of degree three. In preparation

  • Li X, Zhou S (2012) Labeling outerplanar graphs with maximum degree three. Preprint

  • Liu DD, Zhu X (2005) Circular distance two labelling and the λ-number for outerplanar graphs. SIAM J Discrete Math 19:281–291

    Article  MathSciNet  MATH  Google Scholar 

  • Marušič D, Pisanski T (2000) Symmetries of hexagonal molecular graphs on the torus. Croat Chem 73:969–981

    Google Scholar 

  • Molloy M, Salavatipour MR (2005) A bound on the chromatic number of the square of a planar graph. J Comb Theory, Ser B 94:189–213

    Article  MathSciNet  MATH  Google Scholar 

  • Sakai D (1994) Labelling chordal graphs: distance two condition. SIAM J Discrete Math 7:133–140

    Article  MathSciNet  MATH  Google Scholar 

  • Stojmenovic I (1997) Honeycomb networks: topological properties and communication algorithms. IEEE Trans Parallel Distrib Syst 8:1036–1042

    Article  Google Scholar 

  • Xiao W, Parhami B (2007) Further mathematical properties of Cayley digraphs applied to hexagonal and honeycomb meshes. Discrete Appl Math 155:1752–1760

    Article  MathSciNet  MATH  Google Scholar 

  • Yang X, Evans DJ, Lai H, Megson GM (2004) Generalized honeycomb torus is hamiltonian. Inf Process Lett 92:31–37

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou S (2004) A channel assignment problem for optical networks modelled by Cayley graphs. Theor Comput Sci 310:501–511

    Article  MATH  Google Scholar 

  • Zhou S (2006) Labelling Cayley graphs on Abelian groups. SIAM J Discrete Math 19:985–1003

    Article  MATH  Google Scholar 

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Acknowledgements

We appreciate the referees for their helpful comments. The work was supported by a Discovery Project Grant (DP0558677) of the Australia Research Council. Li was supported by a grant (11171129) of the National Natural Science Foundation of China. Zhou was supported by a Future Fellowship (FT110100629) and a Discovery Project Grant (DP120101081) of the Australian Research Council.

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Authors and Affiliations

  1. Department of Mathematics, Central China Normal University, Wuhan, 430079, China

    Xiangwen Li

  2. School of Information Technology, Deakin University, Burwood, VIC, 3125, Australia

    Vicky Mak-Hau

  3. Department of Mathematics and Statistics, The University of Melbourne, Parkville, VIC, 3010, Australia

    Sanming Zhou

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  1. Xiangwen Li
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  2. Vicky Mak-Hau
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  3. Sanming Zhou
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Correspondence to Sanming Zhou.

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Li, X., Mak-Hau, V. & Zhou, S. The L(2,1)-labelling problem for cubic Cayley graphs on dihedral groups. J Comb Optim 25, 716–736 (2013). https://doi.org/10.1007/s10878-012-9525-4

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  • DOI: https://doi.org/10.1007/s10878-012-9525-4

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