  It was discovered a few years ago that many networks in the real world
exhibit self-similarity. A lot of researches on the structures and processes on
real and artificial fractal complex networks have been done, drawing an analogy
to critical phenomena. However, the non-Markovian dynamics on fractal networks
has not been understood well yet. We here study the self-avoiding walk on
complex fractal networks through the mapping of the self-avoiding walk to the
n-vector model by a generating function formalism. First, we analytically
calculate the critical exponent {\nu} and the effective coordination number
(the connective constant) by a renormalization-group analysis in various
fractal dimensions. We find that the exponent {\nu} is equal to the exponent of
displacement, which describes the speed of diffusion in terms of the shortest
distance. Second, by obtaining an exact solution, we present an example which
supports the well-known conjecture that the universality class of the
self-avoiding walk is not determined only by a fractal dimension. Our finding
suggests that the scaling theory of polymers can be applied to graphs which
lack the Euclidian distance as well. Furthermore, the self-avoiding walk has
been exactly solved only on a few lattices embedded in the Euclidian space, but
we show that consideration on general graphs can simplify analytic calculations
and leads to a better understanding of critical phenomena. The scaling theory
of the self-avoiding path will shed light on the relationship between path
numeration problems in graph theory and statistical nature of paths.
