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Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733--748] introduced $e^{^{*}}(G)=|V(G)|(k-1)-{k \choose 2}$ as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319--320] conjectured that any minimal k-UCG contains a Kk as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k-t, for each $t \geq 1$ and each k, when k is large enough. This also... 

Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733-748] introduced e* (G) = |V(G)|(k - 1) - (k2) as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319-320] conjectured that any minimal A-UCG contains a Kk as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k - t, for each t ≥ 1 and each k, when k is large enough. This also improves some known...