  For a class of graphs for which the Ramsey number $R(i,j)$ is upper bounded
by $ci^aj^b$, for some constants $a,b,$ and $c$, it is shown that the clique
covering scheme approximates the broadcast rate of every $n$-node index coding
problem in the class within a multiplicative factor of $c^{\frac{1}{a+b+1}}
n^{\frac{a+b}{a+b+1}}$ for every $n$. Using this theorem and some graph
theoretic arguments, it is demonstrated that the broadcast rate of planar
graphs, line graphs and fuzzy circular interval graphs is approximated by the
clique covering scheme within a factor of $n^{\frac{2}{3}}$.
