Discussion
The upper bound of computing time of m coloring decision problem is
Answer: C
For the m-colouring decision problem on n vertices, a backtracking algorithm tries the m colours for each vertex, so there are up to \(m^{n}\) assignments in the state space, and each one is checked in time proportional to n. So the upper bound of the computing time is \(O(nm^{n})\), option (C). The bounds \(O(nm)\) and \(O(n^{m})\) are polynomial and too small for this exponential search, and the last one is too large.
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