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arXiv · 2601.15520

Colour ratio in Prim's ranking of bipartite graphs

Félix Kahane·Minmin Wang
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Abstract

We consider a complete bipartite graph of size $n$ endowed with i.i.d. uniform edge weights and run Prim's Algorithm to obtain a ranking of its vertices. Let $\rho^{(n)}_k$ be the proportion of black vertices among the first $k$ vertices in this ranking. We characterise the limit behaviour of $\rho^{(n)}_k$ as both $n$ and $k$ tend to infinity. Our results show that in general the limit of $\rho^{(n)}_k$, when existing, differs from the overall proportion of the black vertices in the graph.

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Félix Kahane, Minmin Wang. 2026-01-21. Colour ratio in Prim's ranking of bipartite graphs. https://arxiv.org/abs/2601.15520

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