Abstract: Lin and Zhang first introduced the combinatorial $p$-th Ricci flows and generalized Chow-Luo's classical results on combinatorial Ricci flows (when $p=2$) to any $p>1$. However, their generalization in Euclidean background geometry is incomplete: they only proved the convergence of discrete curvatures along the flow, but failed to establish the convergence of circle packing metrics. In this paper, by introducing a time-dependent normalization term $C(t)$, we overcome the core obstacle arising from the lack of compactness for solutions to combinatorial $p$-th Ricci flows. Our results improve those of Lin and Zhang and generalize Chow-Luo's classical results in Euclidean background geometry from $p=2$ to any $p>1$ in full generality.

Key words: Combinatorial $p$-th Ricci flow, Time-dependent normalization term, Circle packing, Euclidean background geometry, Constant curvature metric