We have known how to get a $2k$ kernel for the Vertex Cover problem for thirty years, and it is not expected to be improved assuming UGC.
My question is, can we do better for planar graphs? It is easy to identify lots of "tight" graphs: a graph has $(2k-1)$ vertices and its minimum Vertex Cover has size $k$. The only known bound is $1.33k$ by Chen et al. [2007 SIAM J. COMPUT.]