On the Color Discrepancy of Spanning Trees in Random and Randomly Perturbed Graphs

TL;DR

This paper investigates the color discrepancy in spanning trees within random and perturbed graphs, revealing that such trees tend to have a significant imbalance in edge color distribution under any 2-coloring.

Contribution

It establishes the existence of spanning trees with large color discrepancy in Erdős-Rényi random graphs and perturbed dense graphs, extending previous understanding of graph coloring properties.

Findings

In Erdős-Rényi graphs above the connectivity threshold, spanning trees with many leaves have a dominant color class.

Adding a few random edges to dense graphs often results in spanning trees with large color discrepancy.

The results hold with high probability for the considered classes of graphs.

Abstract

In this work, we study the color discrepancy of spanning trees in random graphs. We show that for the Erd\H{o}s-R\'enyi 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.