Discrepancies of Spanning Trees and Hamilton Cycles

TL;DR

This paper investigates the multicolour discrepancy of spanning trees and Hamilton cycles in graphs, providing new bounds, generalizations of recent results, and exact asymptotics for graphs with expansion properties.

Contribution

It establishes a fundamental relation between spanning-tree discrepancy and vertex separation, generalizes recent results to all graphs and colours, and determines the discrepancy for the hypercube.

Findings

Spanning-tree discrepancy equals vertex separation measure up to a constant.

Extended discrepancy bounds to graphs with expansion properties.

Determined Hamilton cycle discrepancy in graphs with large minimum degree.

Abstract

We study the multicolour discrepancy of spanning trees and Hamilton cycles in graphs. As our main result, we show that under very mild conditions, the $r$-colour spanning-tree discrepancy of a graph $G$ is equal, up to a constant, to the minimum $s$ such that $G$ can be separated into $r$ equal parts by deleting $s$ vertices. This result arguably resolves the question of estimating the spanning-tree discrepancy in essentially all graphs of interest. In particular, it allows us to immediately deduce as corollaries most of the results that appear in a recent paper of Balogh, Csaba, Jing and Pluh\'{a}r, proving them in wider generality and for any number of colours. We also obtain several new results, such as determining the spanning-tree discrepancy of the hypercube. For the special case of graphs possessing certain expansion properties, we obtain exact asymptotic bounds. We also study the multicolour discrepancy of Hamilton cycles in graphs of large minimum degree, showing that in any $r$-colouring of the edges of a graph with $n$ vertices and minimum degree at least $\frac{r+1}{2r}n + d$, there must exist a Hamilton cycle with at least $\frac{n}{r} + 2d$ edges in some colour. This extends a result of Balogh et al., who established the case $r = 2$. The constant $\frac{r+1}{2r}$ in this result is optimal; it cannot be replaced by any smaller constant.