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Let $G$ be a simple undirected uniquely vertex $k$-colorable graph, or a $k$-UCG for short. M. Truszczynski [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 contains a $K_k$ 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 improves some known results for the case $t=1$.
Second, we analyze the parameter $\Lambda (G)=|E(G)|-e^*(G)$ for our constructions, and we obtain some bounds for the functions
$$\lambda_t(k)=\min\{\Lambda(G): G\mathrm{is \ a} k-\mathrm{UCG \ and} cl(G)=k-t\},$$
$$\nu_t(k)=\min\{|V(G)|: G\mathrm{is \ a} k-\mathrm{UCG \ and} cl(G)=k-t\}.$$
Also, we introduce some possible applications of the technique in cryptography and data compression.
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