关于奇环图（snarks）的最小顶点覆盖
On the minimum vertex cover of snarks
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中文总结 AI 辅助
本研究针对奇环图，证明判定其是否存在指定大小顶点覆盖为NP完全问题，并确定了多个奇环图子类的顶点覆盖数。
中文摘要 AI 辅助
图$G$的顶点覆盖是顶点集$V(G)$的子集$C \subseteq V(G)$，使得图$G$的每条边都与$C$中至少一个顶点关联。$G$的顶点覆盖数是其最小顶点覆盖的基数，记为$\tau(G)$。奇环图（snark）是连通、无桥的三次图，其边色数为4，即无法仅用3种颜色对边进行恰当着色。本研究针对各类奇环图，探究确定其最小顶点覆盖值的问题：给定正整数$k$，首先证明判定任意奇环图是否存在大小$|C| \leq k$的顶点覆盖$C$是NP完全问题；其次确定了奇环图多个子类的顶点覆盖数$\tau(G)$，包括Flower奇环图、Goldberg奇环图、广义Blanuša奇环图及Loupekine奇环图。
英文摘要
A vertex cover of a graph $G$ is a subset of vertices $C \subseteq V(G)$ such that every edge of $G$ is incident to at least one vertex in $C$. The vertex cover number of $G$ is the minimum cardinality of a vertex cover of $G$ and is denoted by $τ(G)$. A snark is a connected, bridgeless, cubic graph that has an edge chromatic number of four, meaning its edges cannot be properly colored with only three colors. In this work, we investigate the problem of determining the value of a minimum vertex cover for classes of snark graphs. Given a positive integer $k$, we firstly prove that determining whether an arbitrary snark has a vertex cover $C$ with size $|C| \leq k$ is an NP-complete problem. Secondly, we determine the vertex cover number $τ(G)$ for several subclasses of snark graphs, such as Flower snarks, Goldberg snarks, Generalized Blanuša snarks and Loupekine snarks.
