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The Relaxed Edge-Coloring Game and k-Degenerate Graphs

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Abstract

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ−j and d≥2j+2 for 0≤j≤Δ−1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k−1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k−j and d≥h(k,j).

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Authors and Affiliations

  1. Linfield College, McMinnville, OR, 97128, USA

    Charles Dunn & Jennifer Firkins Nordstrom

  2. University of Minnesota, Minneapolis, MN, 55455, USA

    David Morawski

Authors
  1. Charles Dunn
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  2. David Morawski
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  3. Jennifer Firkins Nordstrom
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Correspondence to Charles Dunn.

Additional information

Partially supported by the NSF grant DMS 0649068

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Dunn, C., Morawski, D. & Nordstrom, J.F. The Relaxed Edge-Coloring Game and k-Degenerate Graphs. Order 32, 347–361 (2015). https://doi.org/10.1007/s11083-014-9336-6

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  • DOI: https://doi.org/10.1007/s11083-014-9336-6

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