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On minimum spanning trees for random Euclidean bipartite graphs
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 3 / May 2024
- Published online by Cambridge University Press:
- 30 November 2023, pp. 319-350
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- Article
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We consider the minimum spanning tree problem on a weighted complete bipartite graph
$K_{n_R, n_B}$ whose 
$n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in 
$d$ dimensions and edge weights are the 
$p$-th power of their Euclidean distance, with 
$p\gt 0$. In the large 
$n$ limit with 
$n_R/n \to \alpha _R$ and 
$0\lt \alpha _R\lt 1$, we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on 
$d$ only. Despite this difference, for 
$p\lt d$, we are able to prove that the total edge costs normalized by the rate 
$n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.
