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Optimal strong parity edge-coloring of complete graphs

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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.

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Authors and Affiliations

  1. Computer Science Department, Knox College, 2 East South Street, Galesburg, IL, 61401-4999, USA

    David P. Bunde

  2. Mathematics Department, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801-2975, USA

    Kevin Milans, Douglas B. West & Hehui Wu

Authors
  1. David P. Bunde
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  2. Kevin Milans
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  3. Douglas B. West
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  4. Hehui Wu
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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.

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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

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  • DOI: https://doi.org/10.1007/s00493-008-2364-3

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