  We apply the method of graphical functions that was recently extended to six
dimensions for scalar theories, to $\phi^3$ theory and compute the $\beta$
function, the wave function anomalous dimension as well as the mass anomalous
dimension in the $\overline{\mbox{MS}}$ scheme to five loops. From the results
we derive the corresponding renormalization group functions for the Lee-Yang
edge singularity problem and percolation theory. After determining the
$\varepsilon$ expansions of the respective critical exponents to
$\mathcal{O}(\varepsilon^5)$ we apply recent resummation technology to obtain
improved exponent estimates in 3, 4 and 5 dimensions. These compare favourably
with estimates from fixed dimension numerical techniques and refine the four
loop results. To assist with this comparison we collated a substantial amount
of data from numerical techniques which are included in tables for each
exponent.
