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We have a graph $G$ with $v$ vertices. In $G$ there is at least a maximum clique with $k$ vertices, with $k\lt v$. Is it true that there is at least a vertex of $G$ such that its degree is less than $k-1$?

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The question is not clear. Is this the claim you are considering:

If the largest clique in a graph $G=\langle V,E\rangle$ is of size $k<|V|$, then there exists a vertex $v\in V$ of degree $d\leq k-1$.

If so then the claim is wrong. Let $d\geq 2$ and consider the full bipartite graph $K_{d,d}=\langle V,E\rangle$, where $V=L\cup R$ is a disjoint union of two sets of size $d$, and $E=\{ \{u,v\} : u\in R, v\in L\}$. Clearly, $K_{d,d}$ is $d$-regular, and the largest clique in it is of size $k=2\leq d$.

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  • $\begingroup$ What I mean in the question is what you wrote. Your example is right, therefore the claim is wrong. Thank you for the answer. $\endgroup$ Aug 29, 2023 at 17:37

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