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Parity and strong parity edge-colorings of graphs

Author

Listed:
  • Hsiang-Chun Hsu

    (National Taiwan University)

  • Gerard J. Chang

    (National Taiwan University
    National Taiwan University
    National Center for Theoretical Sciences)

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. A parity edge-coloring (respectively, strong parity edge-coloring) is an edge-coloring in which there is no nontrivial parity path (respectively, open parity walk). The parity edge-chromatic number p(G) (respectively, strong parity edge-chromatic number $\widehat{p}(G)$ ) is the least number of colors in a parity edge-coloring (respectively, strong parity edge-coloring) of G. Notice that $\widehat{p}(G) \ge p(G) \ge \chi'(G) \ge \Delta(G)$ for any graph G. In this paper, we determine $\widehat{p}(G)$ and p(G) for some complete bipartite graphs and some products of graphs. For instance, we determine $\widehat{p}(K_{m,n})$ and p(K m,n ) for m≤n with n≡0,−1,−2 (mod 2⌈lg m⌉).

Suggested Citation

  • Hsiang-Chun Hsu & Gerard J. Chang, 2012. "Parity and strong parity edge-colorings of graphs," Journal of Combinatorial Optimization, Springer, vol. 24(4), pages 427-436, November.
    • Handle: RePEc:spr:jcomop:v:24:y:2012:i:4:d:10.1007_s10878-011-9398-y
    DOI: 10.1007/s10878-011-9398-y
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