  Let $S$ be a finite non-commutative semigroup. The commuting graph of $S$,
denoted $\cg(S)$, is the graph whose vertices are the non-central elements of
$S$ and whose edges are the sets $\{a,b\}$ of vertices such that $a\ne b$ and
$ab=ba$. Denote by $T(X)$ the semigroup of full transformations on a finite set
$X$. Let $J$ be any ideal of $T(X)$ such that $J$ is different from the ideal
of constant transformations on $X$. We prove that if $|X|\geq4$, then, with a
few exceptions, the diameter of $\cg(J)$ is 5. On the other hand, we prove that
for every positive integer $n$, there exists a semigroup $S$ such that the
diameter of $\cg(S)$ is $n$. We also study the left paths in $\cg(S)$, that is,
paths $a_1-a_2-...-a_m$ such that $a_1\ne a_m$ and $a_1a_i=a_ma_i$ for all
$i\in \{1,\ldot, m\}$. We prove that for every positive integer $n\geq2$,
except $n=3$, there exists a semigroup whose shortest left path has length $n$.
As a corollary, we use the previous results to solve a purely algebraic old
problem posed by B.M. Schein.
