# 99 questions/Solutions/86

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## Latest revision as of 20:14, 22 November 2013

(**) Node degree and graph coloration

Use Welch-Powell's algorithm to paint the nodes of a graph in such a way that adjacent nodes have different colors.

data Graph a = Graph [a] [(a, a)] deriving (Show, Eq) data Adjacency a = Adj [(a, [a])] deriving (Show, Eq) petersen = Graph ['a', 'b', 'c', 'd', 'e', 'f', 'g', 'h', 'i', 'j'] [('a', 'b'), ('a', 'e'), ('a', 'f'), ('b', 'c'), ('b', 'g'), ('c', 'd'), ('c', 'h'), ('d', 'e'), ('d', 'i'), ('e', 'j'), ('f', 'h'), ('f', 'i'), ('g', 'i'), ('g', 'j'), ('h', 'j')] -- produces graph coloration using Welch-Powell algorithm kcolor :: (Eq a, Ord a) => Graph a -> [(a, Int)] kcolor g = kcolor' x [] 1 where Adj x = sortg g kcolor' [] ys _ = ys kcolor' xs ys n = let ys' = color xs ys n in kcolor' [x | x <- xs, notElem (fst x, n) ys'] ys' (n + 1) color [] ys n = ys color ((v, e):xs) ys n = if any (\x -> (x, n) `elem` ys) e then color xs ys n else color xs ((v, n) : ys) n -- determines chromatic number, given graph coloration chromatic :: [(a, Int)] -> Int chromatic x = length $ foldr (\(a, n) xs -> if n `elem` xs then xs else n : xs) [] x -- converts from graph to adjacency matrix representations graphToAdj :: (Eq a) => Graph a -> Adjacency a graphToAdj (Graph [] _) = Adj [] graphToAdj (Graph (x:xs) ys) = Adj ((x, ys >>= f) : zs) where f (a, b) | a == x = [b] | b == x = [a] | otherwise = [] Adj zs = graphToAdj (Graph xs ys) -- produces graph sorted by node degree sortg :: (Eq a, Ord a) => Graph a -> Adjacency a sortg g = Adj $ map (\(a, b) -> (a, sort b 1 maximum)) $ sort x 1 maxv where Adj x = graphToAdj g sort [] _ _ = [] sort xs n f = let m = f xs in m : sort [x | x <- xs, x /= m] (n + 1) f maxv (x:xs) = foldr (\a@(a1, _) b@(b1, _) -> if a1 > b1 then a else b) x xs