不含长度模3为0或模6为4的环的图
On graphs without cycles of length $0$ modulo $3$ or $4$ modulo $6$
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中文总结 AI 辅助
该研究确定了不含长度模3为0或模6为4的环的n顶点图的精确最大边数，为(11/8)n−7/4，且构造了达到该上界的图类。
中文摘要 AI 辅助
我们研究不含长度可被3整除或模6余4的环的图。证明每个n顶点图G（n≥2）满足边数e(G)≤(11/8)n−7/4，当且仅当n=8k+2（k为非负整数）且G同于显式构造的图H_k时等号成立。还对每个n≥2构造了一个n顶点图，其边数为⌊(11/8)n−7/4⌋，满足相同环限制，因此这是所有n≥2时的精确最大边数。
英文摘要
We study graphs containing no cycle whose length is divisible by $3$ or congruent to $4$ modulo $6$. We prove that every such $n$-vertex graph $G$, where $n \ge 2$, satisfies $e(G) \le (11/8)n-7/4$. Moreover, equality holds if and only if $n=8k+2$ for some nonnegative integer $k$ and $G$ is isomorphic to the explicitly constructed graph $H_k$. We also construct, for every $n\geq2$, an $n$-vertex graph with $\left\lfloor (11/8)n-7/4 \right\rfloor$ edges satisfying the same cycle restriction. Consequently, this is the exact maximum number of edges for every $n \ge 2$.
