We study a phenomenon consisting in the fact that many problems
about graphs is enough to solve with certain errors if their number
grows asymptotically slower than the order of graphs.  For example,
the four color theorem is equivalent with the statement that the
vertices of every $5$-connected planar graph of order $n$ can be
assigned four numbers so that at most $b_n$ couples of neighboring
vertices receive the same number, where $\{ b_n\}$ is a sequence of
nonnegative integers satisfying $\liminf_{n\to\infty}b_n/n=0$. A
similar principle holds for the Thomassen's conjecture on
hamiltonian cycles in 4-connected line graphs, the conjecture of
Fleischner suggesting that every cyclically 4-edge-connected cubic
graph has either a 3-edge-coloring or a dominating cycle, the
nowhere-zero $5$- and $3$-flow problems of Tutte and the circular
flow conjecture of Jaeger.  Furthermore, the last two problems
remain to verify for hamiltonian graphs.
