Almost colour-balanced spanning forests in complete graphs
Lawrence Hollom, Adva Mond, and Julien Portier

TL;DR
This paper proves that in a balanced two-coloured complete graph, one can embed any forest with bounded maximum degree with a small imbalance between red and blue edges, nearly optimal and resolving a previous conjecture.
Contribution
It establishes near-optimal bounds on the imbalance for embedding forests with bounded degree into balanced two-coloured complete graphs, resolving a conjecture.
Findings
Imbalance at most Δ/2 + 18 for forests with maximum degree Δ.
Tighter bounds for small Δ, constant imbalance when Δ < n(1/4 - η).
Resolution of a conjecture by Mohr, Pardey, and Rautenbach.
Abstract
Given whose edges are coloured red and blue, and a forest of order , we seek embeddings of with small imbalance, that is, difference between the numbers of red and blue edges. We show that if the -colouring of the edges of is balanced, meaning that the numbers of red and blue edges are equal, and has maximum degree , then one can find an embedding of into whose imbalance is at most , which is essentially best possible and resolves a conjecture of Mohr, Pardey, and Rautenbach. Furthermore, we give a tighter bound for the imbalance for small values of . In particular, we prove that the imbalance can be taken to be constant in the case where for any constant .
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
