AI 中文总结
本文针对立方图中(1,2)-控制的两个猜想，找到非三叶虫图的反例并构造无限反例族，证明两猜想不成立，不影响Henning等人的相关猜想。
AI 中文摘要
设G是阶为n的立方图，诱导圈顶点数cind(G)是G中诱导出2-正则子图的顶点集的最大规模；γ_{1,2}(G)表示G的(1,2)-控制数。Erves和Tepeh引入了三叶虫图T_n，满足γ_{1,2}(T_n) > cind(T_n)，并提出两个猜想：第一个是每个满足cind(G) ≥ n/2 + 2的立方图G都有γ_{1,2}(G) ≤ cind(G)；第二个是连通立方图G满足γ_{1,2}(G) > cind(G)当且仅当G是三叶虫图。本文证明这两个猜想均不成立，通过计算机搜索找到n=18、20、22时非三叶虫图的反例，还构造了阶为n=20+4k的无限族H(k)，对每个k≥0，证明cind(H(k))=n/2 + 2且γ_{1,2}(H(k))=n/2 + 3，因此两个猜想在无限多阶n下均不成立，本文的例子不影响Henning等人提出的“每个立方图G都有cind(G)≥n/2”的猜想，该猜想仍未解决。
英文摘要
Let G be a cubic graph of order n. The induced cycles vertex number cind(G) is the largest size of a vertex set that induces a 2-regular subgraph of G. By gamma_1,2(G) we denote the (1,2)-domination number of G. Erves and Tepeh introduced the trilobite graphs T_n, which satisfy gamma_1,2(T_n) > cind(T_n). They stated two conjectures, the first of which says that every cubic graph G with cind(G) >= n/2 + 2 satisfies gamma_1,2(G) <= cind(G). The second says that a connected cubic graph G satisfies gamma_1,2(G) > cind(G) if and only if G is a trilobite. We show that both conjectures are false. A computer search finds counterexamples that are not trilobites already for n = 18, 20 and 22. We also construct an infinite family H(k) of order n = 20 + 4k. For every k >= 0 we prove that cind(H(k)) = n/2 + 2 and gamma_1,2(H(k)) = n/2 + 3. Hence both conjectures fail for infinitely many orders n.
Comments15 pages, 1 figure
