The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs G, H, and lists \(L(v)\subseteq V(H)\) for every \(v\in V(G)\), a list homomorphism is a function \(f:V(G)\rightarrow V(H)\) that preserves the edges (i.e., \(uv\in E(G)\) implies \(f(u)f(v)\in E(H)\)) and respects the lists (i.e., \(f(v)\in L(v))\). Standard techniques show that if G is given with a tree decomposition of width t, then the number of list homomorphisms can be counted in time \(|V(H)|^t\cdot n^{\mathcal {O}(1)}\). Our main result is determining, for every fixed graph H, how much the base \(|V(H)|\) in the running time can be improved. For a connected graph H, we define irr(H) in the following way: if H has a loop or is nonbipartite, then irr(H) is the maximum size of a set \(S\subseteq V(H)\) where any two vertices have different neighborhoods; if H is bipartite, then irr(H) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected H, we define irr(H) as the maximum of irr(C) over every connected component C of H. It follows from earlier results that if irr(H)=1, then the problem of counting list homomorphisms to H is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph H, the number of list homomorphisms from (G,L) to H

这项工作的目标是对有界树宽图上的一系列广义着色问题（列表同态）的计数复杂性给出精确的界限。给定图G、H和对于每个v ∈ V ( G ) 的列表L ( v )⊆ V ( H ) ，列表同态是保留边的函数f : V ( G ) → V ( H ) （即， uv ε E ( G ) 意味着f ( u ) f ( v ) ε E ( H )) 并遵守列表（即f ( v ) ε L ( v )）。标准技术表明，如果G给出宽度为t 的树分解，则可以在时间 \(|V(H)|^t\cdot n^{\mathcal {O}(1) 内计算列表同态的数量} \)。我们的主要结果是确定对于每个固定图H，基数 | 是多少。V（H）| 在运行时间上可以得到改善。对于连通图H，我们按以下方式定义 \(\operatorname{irr}(H) \)：如果H有环或非二部，则 \(\operatorname{irr}(H) \) 是最大尺寸集合S ⊆ V ( H ) 的任意两个顶点具有不同的邻域；如果H是二分的，则 \(\operatorname{irr}(H) \) 是完全属于二分类之一的集合的最大大小。对于断开的H ，我们将 \(\operatorname{irr}(H) \) 定义为H的每个连通分量C上 \(\operatorname{irr}(C) \) 的最大值。从之前的结果可以看出，如果 \(\operatorname{irr}(H)=1 \)，那么计算H的列表同态的问题是多项式时间可解的，否则是#P-困难的。我们证明，对于每个固定图H ，从 ( G , L ) 到H的列表同态数