The chromatic-number conjecture for preferential attachment graphs

Let $PA_t(m,\delta)$ be the preferential attachment graph at time $t$, with parameters $m$ and $\delta$, and let $(PA_t(m,\delta))_{t\ge 1}$ denote the resulting graph sequence. The chromatic-number conjecture. For every $m\ge 1$ and every $\delta>-m$, the chromatic number of $(PA_t(m,\delta))_{t\ge 1}$ converges almost surely to $m+1$. This extends the proved result for $\delta\in(-m,0)$ and asks whether the same limiting behavior holds throughout the range $\delta>-m$; the case $\tau\ge 3$ is identified as an open direction in the paper.