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It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\\lfloor\\frac{n-r+1}{2}\\rfloor, \\lceil\\frac{n-r+1}{2}\\rceil}$. 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