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Egalitarian Edge Orderings of Complete Graphs

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Abstract

For a consecutive ordering of the edges of a graph \(G=(V,E)\), the point sum of a vertex is the sum of the indices of edges incident with that vertex. Motivated by questions of balancing accesses in data placements in the presence of popularity rankings, an edge ordering is egalitarian when all point sums are equal, and almost egalitarian when two point sums differ by at most 1. It is established herein that complete graphs on n vertices admit an egalitarian edge ordering when \(n \equiv 1,2,3 \pmod {4}\) and \(n \not \in \{3,5\}\), or an almost egalitarian edge ordering when \(n \equiv 0 \pmod {4}\) and \(n \ne 4\).

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References

  1. Chee, Y.M., Colbourn, C.J., Dau, H., Gabrys, R., Ling, A.C.H., Lusi, D., Milenkovic, O.: Access balancing in storage systems by labeling partial Steiner systems. Des. Codes Crypt. 88, 2361–2376 (2020)

    Article  MathSciNet  Google Scholar 

  2. Coahran, M.M., Colbourn, C.J.: Maximum and average access cost in double erasure RAID disk arrays. In: Proceedings of the Thirty-Fifth Southeastern International Conference on Combinatorics, Graph Theory and Computing, vol. 167, pp. 209–219 (2004)

  3. Cohen, M.B., Colbourn, C.J.: Ladder orderings of pairs and RAID performance. Discrete Appl. Math. 138(1–2), 35–46 (2004)

    Article  MathSciNet  Google Scholar 

  4. Cohen, M.B., Colbourn, C.J., Froncek, D.: Cluttered orderings for the complete graph. In: Computing and combinatorics (Guilin, 2001), Lecture Notes in Comput. Sci., vol. 2108, pp. 420–431. Springer, Berlin (2001)

  5. Colbourn, C.J.: Egalitarian Steiner triple systems for data popularity (2020) (Submitted)

  6. Colbourn, C.J.: Popularity block labelling for Steiner systems. In: Seventeenth International Workshop on Algebraic and Combinatorial Coding Theory (ACCT2020), pp. 41–46 (2020)

  7. Dau, H., Milenkovic, O.: MaxMinSum Steiner systems for access balancing in distributed storage. SIAM J. Discrete Math. 32(3), 1644–1671 (2018)

    Article  MathSciNet  Google Scholar 

  8. Dewar, M., Stevens, B.: Ordering block designs: Gray codes, universal cycles and configuration orderings. CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York (2012)

    Book  Google Scholar 

  9. Ho, Y.S., Lee, S.M.: Some initial results on the supermagicness of regular complete \(k\)-partite graphs. J. Combin. Math. Combin. Comput. 39, 3–17 (2001)

    MathSciNet  MATH  Google Scholar 

  10. Stewart, B.M.: Supermagic complete graphs. Can. J. Math. 19, 427–438 (1967)

    Article  MathSciNet  Google Scholar 

  11. Yu, W., Zhang, X., Ge, G.: Optimal fraction repetition codes for access-balancing in distributed storage (2019). arXiv:1912.07779

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Acknowledgements

The work was supported by NSF Grant CCF 1816913. Thanks to Yeow Meng Chee, Dylan Lusi, and Olgica Milenkovic for helpful discussions. Thanks also to three anonymous referees for excellent comments and corrections.

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The work was supported by NSF Grant CCF 1816913.

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Correspondence to Charles J. Colbourn.

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Colbourn, C.J. Egalitarian Edge Orderings of Complete Graphs. Graphs and Combinatorics 37, 1405–1413 (2021). https://doi.org/10.1007/s00373-021-02326-5

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