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The Hadwiger Conjecture corresponds to the case $\\alpha=1$, $\\beta=-1$ and $N_t=1$. Kawarabayashi and Mohar [\\emph{J. Combin. Theory Ser. B}, 2007] proved this relaxation with $\\alpha={31/2}$ and $\\beta=0$ (and $N_t$ a huge function of $t$). 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