On July 21, 2020, Ilkyoo Choi (최일규) from Hankuk University of Foreign Studies gave a talk on the flexibility of planar graphs, a concept related to extending a a precoloring at the Discrete Math Seminar. The title of his talk was “Flexibility of Planar Graphs“.
Oftentimes in chromatic graph theory, precoloring techniques are utilized in order to obtain the desired coloring result. For example, Thomassen’s proof for 5-choosability of planar graphs actually shows that two adjacent vertices on the same face can be precolored. In this vein, we investigate a precoloring extension problem formalized by Dvorak, Norin, and Postle named flexibility. Given a list assignment $L$ on a graph $G$, an $L$-request is a function on a subset $S$ of the vertices that indicates a preferred color in $L(v)$ for each vertex $v\in S$. A graph $G$ is $\varepsilon$-flexible for list size $k$ if given a $k$-list assignment $L$ and an $L$-request, there is an $L$-coloring of $G$ satisfying an $\varepsilon$-fraction of the requests in $S$. We survey known results regarding this new concept, and prove some new results regarding flexibility of planar graphs.