If we consider the complete bipartite graph $K_{n,m},$ where $n,m$ are natural numbers with $n < m,$ then the maximum degree of a vertex in this graph is $=m$ and there are at least $n$ many vertices having degree $m$, and so the number of "kings" (as per the definition above) in this graph is $ = n,$ and the total number of vertices $=n+m.$
Now, since $n,m$ can be anything (with $n \le m$), so one can take $n,m$ to be such that $n/(n+m) > \frac{1}{3},$ and thus in this case we see that the number of kings in this graph is $> \frac{1}{3} \cdot$ the number of vertices in the graph.
For such graphs you can just set $n=m-1$ to see that you cannot get an "asymptotic" bound of $< 1/2.$