# Subgraph Isomophism Problem - Color Coding Technique - Proof Sketch

I am reading the paper Color Coding by Alon, Yuster, and Zwick. They state a theorem (6.3) that says if $H$ is a graph on $k$ vertices with treewidth $t$ and $G = (V, E)$, then a subgraph of $G$ isomorphic to $H$ can be found in $2^{O(k)}V^{t+1}$ expected time. They do not include a proof, but do state that the proof is similar to the case where $H$ is a forest (namely theorem 6.1, which they do provide a proof of).

As I am trying to understand how they algorithm would work for graphs of bounded treewidth, I was wondering if anyone could provide a proof sketch of the algorithm, as I don't see how it would be similar to the proof of 6.1 - Any help would really be appreciated, as this is for a course project, and I am having difficulties figuring out the algorithm.

A copy of the paper can be found here: http://www.tau.ac.il/~nogaa/PDFS/col5.pdf

The idea is to repeat the proof of Theorem 6.1, only during the recursive construction in the second paragraph, we use the tree decomposition of $H$ rather than $H$ itself. We pick a vertex in the tree decomposition, which correspond to up to $t+1$ vertices of $H$, and split $H$ into two parts $H',H''$ accordingly, applying the algorithm recursively. Since we are dealing with groups of up to $t+1$ vertices at a time, we have to calculate the color sets for sets of $t+1$ vertices rather than for single vertices.
• Could you provide a bit more information as to how the colorsets for $t + 1$ vertices are computed? Is that what correspondents to the portion $V^{t+1}$ in the runtime?
• I'm afraid you'll have to work this out yourself. The $V^{t+1}$ factor indeed comes from the fact that we have to assign colorsets to ordered sets of $t+1$ vertices. Mar 19 '14 at 2:36