We investigate a competitive version of the coloring number of a graph G = (V,
E). For a fixed linear ordering L of V let s (L) be one more than the maximum
outdegree of G when G is oriented so that x ← y if x < L y. The coloring number
of G is the minimum of s (L) over all such orderings. The (a, b)-marking game is
played on a graph G = (V, E) as follows. At the start all vertices are unmarked.
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