Flexibility of Planar Graphs -- Sharpening the Tools to Get Lists of Size Four

A graph where each vertex \(v\) has a list \(L(v)\) of available colors is \(L\)-colorable if there is a proper coloring such that the color of \(v\) is in \(L(v)\) for each \(v\). A graph is \(k\)-choosable if every assignment \(L\) of at least \(k\) colors to each vertex guarantees an \(L\)-coloring. Given a list assignment \(L\), an \(L\)-request for a vertex \(v\) is a color \(c\in L(v)\). In this paper, we look at a variant of the widely studied class of precoloring extension problems from [Z. Dvořák, S. Norin, and L. Postle: List coloring with requests. J. Graph Theory 2019], wherein one must satisfy "enough", as opposed to all, of the requested set of precolors. A graph \(G\) is \(\varepsilon\)-flexible for list size \(k\) if for any \(k\)-list assignment \(L\), and any set \(S\) of \(L\)-requests, there is an \(L\)-coloring of \(G\) satisfying an \(\varepsilon\)-fraction of the requests in \(S\). It is conjectured that planar graphs are \(\varepsilon\)-flexible for list size \(5\), yet it is proved only for list size \(6\) and for certain subclasses of planar graphs. We give a stronger version of the main tool used in the proofs of the aforementioned results. By doing so, we improve upon a result by Masařík and show that planar graphs without \(K_4^-\) are \(\varepsilon\)-flexible for list size \(5\). We also prove that planar graphs without \(4\)-cycles and \(3\)-cycle distance at least 2 are \(\varepsilon\)-flexible for list size \(4\). Finally, we introduce a new (slightly weaker) form of \(\varepsilon\)-flexibility where each vertex has exactly one request. In that setting, we provide a stronger tool and we demonstrate its usefulness to further extend the class of graphs that are \(\varepsilon\)-flexible for list size \(5\).