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 Pluhár, 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.

我们研究了图中生成树和汉密尔顿循环的多色差异。作为我们的主要结果，我们表明在非常温和的条件下，图G的r颜色生成树差异等于（直到一个常数）到最小值s，这样G可以通过删除s分成r个相等的部分顶点。这个结果可以说解决了在基本上所有感兴趣的图中估计生成树差异的问题。特别是，它使我们能够立即将出现在 Balogh、Csaba、Jing 和 Pluhár 的最新论文中的大多数结果作为推论推论出来，并以更广泛的普遍性和任意数量的颜色证明它们。我们还获得了几个新结果，例如确定超立方体的生成树差异。对于具有某些扩展属性的图的特殊情况，我们获得了精确的渐近界。

我们还研究了大最小度数图中 Hamilton 循环的多色差异，表明在具有n个顶点和最小度数的图的边的任何r着色中，至少$\frac{r+1}{2r}n+d$, 必须存在至少有一个 Hamilton 圈$\frac{n}{r}+2d$一些颜色的边缘。这扩展了 Balogh 等人的结果，他们确立了这个案例$r=2$. 常数$\frac{r+1}{2r}$在这个结果中是最优的；它不能被任何更小的常数代替。