A graph homomorphism is a vertex map which carries edges from a source graph to edges in a target graph. The instances of the Weighted Maximum H-Colourable Subgraph problem (MAX H-COL) are edge-weighted graphs G and the objective is to find a subgraph of G that has maximal total edge weight, under the condition that the subgraph has a homomorphism to H; note that for H=K_k this problem is equivalent to MAX k-CUT. Farnqvist et al. have introduced a parameter on the space of graphs that allows close study of the approximability properties of MAX H-COL. Specifically, it can be used to extend previously known (in)approximability results to larger classes of graphs. Here, we investigate the properties of this parameter on circular complete graphs K_{p/q}, where 2 <= p/q <= 3. The results are extended to K_4-minor-free graphs and graphs with bounded maximum average degree. We also consider connections with Samal's work on fractional covering by cuts: we address, and decide, two conjectures concerning cubical chromatic numbers.
许多重要的图论概念可以编码为计数图同态问题，例如统计物理中的配分函数，特别是独立集和着色。在这篇文章中，我们研究了$\#_p\mathsf{H}$ OMS $\mathsf{T}$ O $H$的复杂性，从输入图到图$H$模素数$p$的图同态计数问题。Dyer和Greenhill证明了一个二分法，说明非模计数图同态的易处理性取决于输入图的结构。由于取消现象的普遍存在，非模计数中许多难以处理的情况在模计数中变得容易处理。然而，在随后的研究计数模$2$的影响，结构的$H$上的易处理性，被证明是持久的，产生类似的二分法。我们的主要结果表明，对于每一棵树$H$和每一个素数$p$，问题$\#_p\mathsf{H}$ OMS $\mathsf{T}$ O $H$是多项式时间可计算的或$\#_p\mathsf{P}$ -完全的.这解决了Faben和Jerrum在模2计数时对每个图$H$陈述这种二分法的猜想。为了证明这个结果，我们研究了同态的结构性质。作为一个重要的过渡，这项研究产生了一个二分法的问题的加权计数独立集的二分图模一些素数$p$。我们的结果是第一次表明，这样的二分法不仅适用于1位功能的模~2的情况下，但所有素数$p$的模计数功能。
