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