Colour ratio in Prim's ranking of bipartite graphs

TL;DR

This paper analyzes the limiting behavior of the proportion of black vertices in the ranking produced by Prim's Algorithm on large bipartite graphs with random edge weights, revealing that the local proportion can differ from the overall graph proportion.

Contribution

It characterizes the asymptotic behavior of the black vertex ratio in Prim's ranking on bipartite graphs, highlighting differences from the global black vertex proportion.

Findings

Limit of black vertex proportion can differ from overall proportion.

Results depend on both graph size and ranking position.

Provides a detailed asymptotic analysis of Prim's Algorithm on bipartite graphs.

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.