The rainbow number ${\rm rb}(G, H)$ is the minimum number of colors $k$ for which any edge-coloring of $G$ with at least $k$ colors guarantees a rainbow subgraph isomorphic to $H$. The rainbow number has many applications in diverse fields such as wireless communication networks, cryptography, bioinformatics, and social network analysis. In this paper, we determine the exact rainbow number $\mathrm{rb}(G, H)$ where $G$ is a multi-hubbed wheel graph $W_d(s)$, defined as the join of $s$ isolated vertices and a cycle $C_d$ of length $d$ (i.e., $W_d(s) = \overline{K_s} + C_d$), and $H = θ_{t,\ell}$ represents a cycle $C_t$ of length $t$ with $0 \leq \ell \leq t-3$ chords emanating from a common vertex, by establishing \[ {\rm rb}(W_{d}(s), θ_{t,\ell}) = \begin{cases} \left\lfloor \dfrac{2t - 5}{t - 2}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 1 \text{ and } t\ge 4, \\[10pt] \left\lfloor \dfrac{3t-10}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 2\text{ and } t\ge 6,\\[10pt] \left\lfloor \dfrac{(s + 1)t - (3s + 4)}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s \geq 3\text{ and } t\ge 7,\\[10pt] \left\lfloor \dfrac{2t - 7}{t - 3}d \right\rfloor + 1, & \text{if } s = 1 \text{ and } t\ge \max\{5,\ell+4\}, \end{cases} \] when $d\geq 3t-5$, with all bounds for the parameter $t$ presented here being tight. This addresses the problems proposed by Jakhar, Budden, and Moun (2025), which involve investigating the rainbow numbers of large cycles and large chorded cycles in wheel graphs (specifically corresponding to the cases in our framework where $s=1$ and $\ell\in \{0,1\}$). Furthermore, it completely determines the rainbow numbers of cycles of arbitrary length in large wheel graphs, thereby generalizing a result of Lan, Shi, and Song (2019).

翻译：彩虹数 ${\rm rb}(G, H)$ 是指满足如下条件的最小颜色数 $k$：对图 $G$ 进行任意使用至少 $k$ 种颜色的边着色，都保证存在一个与 $H$ 同构的彩虹子图。彩虹数在无线通信网络、密码学、生物信息学和社会网络分析等多个领域具有广泛应用。本文确定了确切彩虹数 $\mathrm{rb}(G, H)$，其中 $G$ 是多辐轮图 $W_d(s)$（定义为 $s$ 个孤立顶点与一个长度为 $d$ 的圈 $C_d$ 的联图，即 $W_d(s) = \overline{K_s} + C_d$），而 $H = θ_{t,\ell}$ 表示一个长度为 $t$ 的圈 $C_t$，且该圈有一个公共顶点引出 $0 \leq \ell \leq t-3$ 条弦。通过建立公式 \[ {\rm rb}(W_{d}(s), θ_{t,\ell}) = \begin{cases} \left\lfloor \dfrac{2t - 5}{t - 2}d \right\rfloor + 1, & \text{若 } \ell=t-3,~s = 1 \text{ 且 } t\ge 4, \\[10pt] \left\lfloor \dfrac{3t-10}{t - 3}d \right\rfloor + 1, & \text{若 } \ell=t-3,~s = 2\text{ 且 } t\ge 6,\\[10pt] \left\lfloor \dfrac{(s + 1)t - (3s + 4)}{t - 3}d \right\rfloor + 1, & \text{若 } \ell=t-3,~s \geq 3\text{ 且 } t\ge 7,\\[10pt] \left\lfloor \dfrac{2t - 7}{t - 3}d \right\rfloor + 1, & \text{若 } s = 1 \text{ 且 } t\ge \max\{5,\ell+4\}, \end{cases} \] 当 $d\geq 3t-5$ 时，其中给出的参数 $t$ 的所有边界都是紧的。这解决了 Jakhar、Budden 和 Moun (2025) 提出的问题，即研究轮图中大圈和大带弦圈的彩虹数（具体对应于我们框架中 $s=1$ 且 $\ell\in \{0,1\}$ 的情形）。此外，该结果完全确定了大型轮图中任意长度圈的彩虹数，从而推广了 Lan、Shi 和 Song (2019) 的一个结果。