AI 中文总结
研究关于图的色数、团数和零强制数的关系，通过无三角形诱导子图建立零强制数下界，提出\((χ, ω, Z)\)猜想，证明其对无三角形正则图成立并提供数值证据。
AI 中文摘要
设\(G\)为具有色数\(\chi(G)\)、团数\(\omega(G)\)和零强制数\(Z(G)\)的图。我们根据无三角形诱导子图建立了\(Z(G)\)的新下界。特别地，我们表明若图\(G\)包含最小度\(\delta(H) \geq 3\)的无三角形诱导子图\(H\)，则\(Z(G)\geq\delta(H)+1\)。受此结果及Taklimi（2013）的界\(\chi(G) \leq Z(G) + 1\)启发，我们猜想\(\chi(G) \leq \left \lceil \frac{\omega(G)+Z(G)+1}{2}\right\rceil\)。作为支持证据，我们证明该猜想对无三角形正则图成立并提供了数值证据。
英文摘要
Let $G$ be a graph with chromatic number $χ(G)$, clique number $ω(G)$ and zero forcing number $Z(G)$. We establish new lower bounds on $Z(G)$ in terms of induced triangle-free subgraphs. In particular, we show that if a graph $G$ contains an induced triangle-free subgraph $H$ with minimum degree $δ(H) \ge 3$, then $Z(G)\geδ(H)+1$. As consequences, we prove that every triangle-free graph satisfies $χ(G)\le\max\{3,Z(G)\}$ and obtain an application to planar graphs. Moreover, we prove that $χ(G)\le \frac{Z(G)}{2}+2$ for every triangle-free graph.
