小反射和二面体拉姆齐数的计算
Computation of small reflective and dihedral Ramsey numbers
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中文总结 AI 辅助
研究小反射和二面体拉姆齐数的计算问题，应用基于SAT的方法，利用Kissat SAT求解器，针对特定图类得到小反射和二面体拉姆齐数的精确值与下界，并得出一般结果和提出猜想。
中文摘要 AI 辅助
本文研究的所有图都是简单、有限的，顶点集为{0, 1, 2, …, n - 1}（n∈N）。对于图G、H以及H顶点集上的置换群Γ，若存在形如ψ∘φ的图同态（其中φ∈Γ且ψ是递增单射），则称H可Γ嵌入G。近期通过引入置换拉姆齐数统一了标准和有序拉姆齐数。本文关注反射（或二面体）拉姆齐数，它是置换拉姆齐数的特定类型，其中每个Γj是Hj自然序顶点集上的反射群（或二面体群）。在双色情况下，应用基于SAT的方法，利用Kissat SAT求解器得到小反射和二面体拉姆齐数的精确值和下界，这些数的两个参数属于单调和交替路径、单调循环、起始中心星、完全图和嵌套匹配等图类。还得出一些一般结果并基于计算结果提出猜想。
英文摘要
Throughout, all graphs are simple, finite and have vertex sets of the form $\{ 0, 1, 2, \ldots, n - 1 \}$ for some $n \in \mathbb{N}$. For graphs $G$ and $H$, and a permutation group $Γ$ on the vertex set of $H$, we say that $H$ is $Γ$-embeddable in $G$ if there exists a graph homomorphism from $H$ to $G$ of the form $ψ\circ φ$, where $φ\in Γ$ and $ψ$ is an increasing injection. Recently, standard and ordered Ramsey numbers of graphs were unified through the introduction of permutational Ramsey numbers, defined as follows. For graphs $H_1, H_2, \ldots, H_k$ and permutation groups $Γ_1, Γ_2, \ldots, Γ_k$ on their respective vertex sets, the permutational Ramsey number $R(H_1^{Γ_1}, H_2^{Γ_2}, \ldots, H_k^{Γ_k})$ is the minimum $n \in \mathbb{N}$ such that for every $k$-edge-coloring of a complete graph on $n$ vertices, there exists some $j \in \{1, 2, \ldots, k\}$ for which $H_j$ is $Γ_j$-embeddable in the spanning subgraph of the complete graph comprising the edges of color $j$. Here, we consider reflective (resp. dihedral) Ramsey numbers, which are a specific class of permutational Ramsey numbers in which each group $Γ_j$ is the reflection group (resp. dihedral group) on the naturally ordered vertex set of $H_j$. Focusing on the two-color case, we apply the SAT-based approach originally proposed by Poljak for ordered Ramsey numbers and recently extended to cyclic Ramsey numbers. We utilize the Kissat SAT solver to obtain exact values and lower bounds for small reflective and dihedral Ramsey numbers whose two arguments belong to the following graph classes: monotone and alternating paths, monotone cycles, start-central stars, complete graphs and nested matchings. We also derive several general results and formulate conjectures based on the computational findings.
