# Breaking frustrated loops in list coloring problem

Suppose we have a graph with $n$ vertices and $n$ lists of colors such that no vertex coloring of the graph using colors from given lists is a proper list coloring. We can characterize the obstacle to proper list coloring in terms of frustrated cycles -- simple cycles such that every list coloring of the induced subgraph is improper.

Given graph and set of color lists I'd like to delete a small number of vertices to break a large number of frustrated cycles.

It sounds like you are asking if the Erdős–Pósa property holds for frustrated cycles. That is, does there exists a function $f: \mathbb{N} \to \mathbb{N}$, such that every graph (with lists) either has $k$ disjoint frustrated cycles, or a set of vertices $X$, of size at most $f(k)$, such that $G -X$ contains no frustrated cycles.