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In this paper we first generalize a classical result of $B$. Toft
(1974) on $r$-type-constructions for graphs (rather than
hypergraphs) and then we show how the result can be used to
construct colour-critical graphs with a special focus on
$\Delta$-colour-critical graphs. This generalization covers most
of known constructions which generate small critical graphs. We
also obtain some upper bounds for the {\it minimum excess
function} $\eta(k,p)$ when $4\leq k \leq 6$; where
$$\eta(k,p)=\min_{G\in K(k,p)} \epsilon (G),$$ in which $\epsilon
(G)=2|E(G)|-|V(G)|(k-1)$, and $K(k,p)$ is the class of all
$k$-colour-critical graphs on $p$ vertices with $\Delta=k$. We use
the techniques to construct an infinite family of
$\Delta$-colour-critical graphs for $\Delta=5$ with a relatively
small minimum excess function; and we prove that $\eta(6,6n)\leq
6(n-1)(n\geq 2)$ which shows that there exists an infinite family
of $\Delta$-colour-critical graphs for $\Delta=6$.
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