# Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

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Every edge is in a red ellipse (the bags). The bags are connected in a tree. The bags an original vertex is part of are connected.

The root

Directed Weighted w3 w1 w2 w5 w9 w11 w8 w10 w7 w6 w4

 Recursively build paths: O(n k )  Can there be algorithms with runtimes on the form f(k)n O(1) ? … and if so, how small can f(k) be?

Consider regular graphs of degree d: Either d>k: There must be k-path. Or d<=k: We can list all potential k-paths in nd k <=nk k time.

<k? B1 B2 B3 B4 B5 B6 Depth First Search

B1B2B3 B4 B5 B6

13 5 246 Prob[rainbow k-path] >= k!/k k ~ e -k

 Dynamic programming over color subsets.  Let D(X,v,k) be True iff there is a path of length k ending in vertex v whose vertices are colored as X.

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