Optimal strong parity edge-coloring of complete graphs

Abstract

A parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let \( \hat p \)(G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always \( \hat p \)(G) ≥ p(G) ≥ χ′(G). We prove that \( \hat p \)(K n ) = 2⌈lgn − 1 for all n. The optimal strong parity edge-coloring of K n is unique when n is a power of 2, and the optimal colorings are completely described for all n.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon: Combinatorial Nullstellensatz, Combin., Prob., Comput. 8 (1999), 7–29.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    N. Alon, J. Grytczuk, M. Hałuszczak and O. Riordan: Nonrepetitive colorings of graphs, Random Structures Algorithms 21 (2002), 336–346.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    B. Bollobás: The art of mathematics. Coffee time in Memphis; Cambridge University Press, New York, 2006, page 186.

    Google Scholar 

  4. [4]

    B. Bollobás and I. Leader: Sums in the grid, Discrete Math. 162 (1996), 31–48.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    D. P. Bunde, K. Milans, D. B. West, and H. Wu: Parity and strong parity edge-coloring of graphs, in Proc. 38th SE Intl. Conf. Comb., Graph Th., Comput., Congressus Numer. 187 (2007), 193–213.

    MATH  MathSciNet  Google Scholar 

  6. [6]

    S. Eliahou and M. Kervaire: Sumsets in vector spaces over finite fields, J. Number Theory 71 (1998), 12–39.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    S. Eliahou and M. Kervaire: Old and new formulas for the Hopf-Stiefel and related functions, Expo. Math. 23 (2005), 127–145.

    MATH  MathSciNet  Google Scholar 

  8. [8]

    I. Havel and J. Morávek: B-valuation of graphs, Czech. Math. J. 22 (1972), 338–351.

    Google Scholar 

  9. [9]

    H. Hopf: Ein topologischer Beitrag zur reellen Algebra, Comment. Math. Helv. 13 (1940/41), 219–239.

    Article  Google Scholar 

  10. [10]

    Gy. Károlyi: A note on the Hopf-Stiefel function, Europ. J. Combin. 27 (2006), 1135–1137.

    MATH  Article  Google Scholar 

  11. [11]

    P. Keevash and B. Sudakov: On a hypergraph Turán problem of Frankl, Combinatorica 25(6) (2005), 673–706.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    A. Plagne: Additive number theory sheds extra light on the Hopf-Stiefel ∘ function, L’Enseignement Math. 49 (2003), 109–116.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    H. Shapiro: The embedding of graphs in cubes and the design of sequential relay circuits, Bell Telephone Laboratories Memorandum, July 1953.

  14. [14]

    E. Stiefel: Über Richtungsfelder in den projektiven Räumen und einen Satz aus der reellen Algebra, Comment. Math. Helv. 13 (1940/41), 201–218.

    Article  MathSciNet  Google Scholar 

  15. [15]

    S. Yuzvinsky: Orthogonal pairings of Euclidean spaces, Michigan Math. J. 28 (1981), 131–145.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Douglas B. West.

Additional information

Partially supported by NSF grant CCR 0093348.

Work supported in part by the NSA under Award No. MDA904-03-1-0037.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bunde, D.P., Milans, K., West, D.B. et al. Optimal strong parity edge-coloring of complete graphs. Combinatorica 28, 625–632 (2008). https://doi.org/10.1007/s00493-008-2364-3

Download citation

Mathematics Subject Classification (2000)

  • 05C15
  • 05C38
  • 11B75