On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs
Wenchong Chen, Xiao-Chuan Liu, Xu Yang
TL;DR
This work investigates color discrepancy in spanning trees within random graphs and randomly perturbed dense graphs. By adopting a hypergraph framework and developing leaf-rich spanning-tree constructions, it proves that in $G \sim G(n,p)$ with $p \ge C/n$ there exists a spanning forest with substantial color imbalance under any 2-edge-coloring, and, in the regime $p = (\log n + \omega(1))/n$, a spanning tree with a linear number of leaves exhibits strong imbalance. The results extend to randomly perturbed dense graphs $G_{\alpha} \cup G(n,p)$, establishing that sparse randomness suffices to enforce large discrepancy in all spanning trees, and they address $r$-colorings as well. Techniques include a leaf-growth strategy via a Leaf-Increasing Algorithm, bipartite matching arguments, and a perturbation-Connectivity framework leading to 3-connectivity, culminating in quantitative lower bounds on the discrepancy $s(H)$. These findings highlight the robustness of color-discrepancy phenomena in network-like structures, with implications for routing and network design in random and perturbed environments.
Abstract
In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erdős-Rényi random graph $G(n,p)$ with $p$ above the connectivity threshold, the following holds with high probability: in every 2-edge-coloring of the graph, there exists a spanning tree with a linear number of leaves such that one color class contains more than $\frac{1 + \varepsilon}{2}n $ of the tree's edges. Here, $\varepsilon>0$ is a small absolute constant independent of $p$. We also extend this line of research to randomly perturbed dense graphs, showing that adding a few random edges to a dense graph typically creates a spanning tree with a large color discrepancy under any 2-edge-coloring.