AI 中文总结
本文针对无爪立方图零强制数上界的三个开放问题展开研究，刻画了满足特定条件的连通无爪立方图，还为具有哈密顿收缩多重图的无爪立方图建立了改进上界，涉及零强制数与图中三角形、菱形数量的关系。
AI 中文摘要
无爪立方图是没有与\(K_{1,3}\)同构的诱导子图的立方图。零强制过程从一组初始着色顶点\(S\)开始。每一步，恰好有一个未着色邻居的着色顶点迫使该邻居着色。若重复此规则能给图\(G\)的每个顶点着色，则\(S\)是零强制集。零强制集的最小基数是零强制数，记为\(Z(G)\)。本文回答了Davila和Henning提出的关于无爪立方图零强制数上界的三个开放问题。刻画了满足\(Z(G)=\alpha(G)+1\)（其中\(\alpha(G)\)是独立数）的连通无爪立方图。此外，对于具有哈密顿收缩多重图的无爪立方图，建立了改进的上界\(Z(G)\leq \frac{T}{2}+D+2\)，其中\(D\)是菱形数量，\(T\)是三角形数量。
英文摘要
A claw-free cubic graph is a cubic graph with no induced subgraph isomorphic to $K_{1,3}$. The zero forcing process begins with an initial set $S$ of colored vertices. At each step, a colored vertex with exactly one uncolored neighbor forces that neighbor to become colored. If repeated applications of this rule color every vertex of $G$, then $S$ is called a zero forcing set. The minimum cardinality of a zero forcing set is the zero forcing number, denoted by $Z(G)$. In this paper, we answer three open questions posed by Davila and Henning concerning upper bounds on the zero forcing number of claw-free cubic graphs. We characterize the connected claw-free cubic graphs satisfying $Z(G)=α(G)+1$, where $α(G)$ is the independence number. In addition, we establish the improved upper bound $Z(G)\leq \frac{T}{2}+D+2$ for claw-free cubic graphs with Hamiltonian contraction multigraphs, where $D$ is the number of diamonds and $T$ is the number of triangles in $G$.
