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Consider the parameter $$\Lambda(G)=\vert E(G)\vert -\vert
V(G)\vert (k-1)+\binom k2$$ for a $k$-chromatic graph $G$, on the
set of vertices $V(G)$ and with the set of edges $E(G)$. It is
known that $\Lambda(G)\geq 0$ for any $k$-chromatic uniquely
vertex-colourable graph $G$ ($k$-UCG), and, S. J. Xu has
conjectured that for any $k$-UCG, $G$, $\Lambda(G)=0$ implies that
${\rm cl}(G)=k$; in which ${\rm cl}(G)$ is the clique number of
$G$. In this paper, first, we introduce the concept of the $core$
of a $k$-UCG as an induced subgraph without any colour-class of
size one, and without any vertex of degree $k-1$. Considering
$(k,n)$-cores as $k$-UCGs on $n$ vertices, we show that
edge-minimal $(k,2k)$-cores do not exist when $k\geq 3$, which
shows that for any edge-minimal $k$-UCG on $2k$ vertices either
the conjecture is true or there exists a colour-class of size one.
Also, we consider the structure of edge-minimal $(k,2k+1)$-cores
and we show that such cores exist for all $k\geq 4$. Moreover, we
characterize all edge-minimal $(4,9)$-cores and we show that there
are only seven such cores (up to isomorphism). Our proof shows
that Xu's conjecture is true in the case of edge-minimal
$(4,9)$-cores.
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