Publication Date



Discrete Mathematics and Combinatorics


Let k be a positive integer, d be a nonnegative integer, and G be a finite graph. Two players, Alice and Bob, play a game on G by coloring the uncolored vertices with colors from a set X of k colors. At all times, the subgraph induced by a color class must have maximum degree at most d. Alice wins the game if all vertices are eventually colored; otherwise, Bob wins. The least k such that Alice has a winning strategy is called the d-relaxed game chromatic number of G, denoted χ gd (G). It is known that there exist graphs such that χ g0 (G) = 3, but χ g1 (G) > 3. We will show that for all positive integers m, there exists a complete multipartite graph G such that m ≤ χ g0 (G) < χ g1 (G).

Document Type

Accepted Version


This article is the author-created version that incorporates referee comments. It is the accepted-for-publication version. The content of this version may be identical to the published version (the version of record) save for value-added elements provided by the publisher (e.g., copy editing, layout changes, or branding consistent with the rest of the publication).


This is an electronic version of an article published in Order, 2012, volume 29, issue 3, pages 507-512. Order is available online at: doi:10.1007/s11083-011-9217-1.

Original Citation

Charles Dunn
Complete multipartite graphs and the relaxed coloring game.
Order, 2012, volume 29, issue 3, pages 507-512



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