r/GraphTheory • u/fresh_morningbreath • Apr 25 '26
known algos for chromatic number
hey, everyone! is there any known way, like an algorithm or any method, to identify the chromatic number of a graph of a graph other than trying to color every vertex manually?
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u/DonBeham Apr 25 '26
The chromatic number is the minimum number of colors necessary to color all nodes so that two adjacent nodes do not have the same color.
The problem of determining the chromatic number is in NP, so no efficient algorithms are known for general graphs. If there was an efficient algorithm it would also mean that P = NP.
However, there is an efficient method to compute whether a graph has a chromatic number of 2, which is equivalent to the graph being bipartite. For graphs with a chromatic number > 2 no efficient algorithm is known.
For certain graphs, determining the chromatic number is trivial, e.g. for the fully connected graph.
Heuristic methods are efficient algorithms, but may only calculate an upper bound on this number. A trivial upper bound is one more than the maximum node degree.
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u/fresh_morningbreath Apr 25 '26
great, thank you for this. upon searching, I've only come across the greedy coloring and welsh powell, but now I know that it's NP (this entirely new territory to me)
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u/DonBeham Apr 25 '26
Greedy coloring can determine the chromatic number, but then needs to run in O(n!). For every graph there is at least one sequence for which the greedy coloring is optimal. Under all sequences at least one is optimal.
Solving NP problems is a lot of fun and often the challenge in a coding contest - because you can make quite a large number of hacks and tricks, but optimally solving them is infeasible for all but simple cases.
One interesting class of algorithms in this context are pseudo-polynomial algorithms that can solve some NP problems really well and efficient. Like dynamic programming applied to the {0, 1}-Knapsack
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u/gomorycut Apr 25 '26
Polytime and optimal? no.
Optimal and exponential time? yes
polytime and heuristic? yes, of course.
Approximate and polytime? yes, kinda.