Exact rainbow numbers of cycle-related graphs in multi-hubbed wheels

Abstract

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 $H = \theta_{t,\ell}$ represents a cycle $C_t$ of length $t$ with $0 \leq \ell \leq t-3$ chords emanating from a common vertex, by establishing \[ {\rm rb}(W_{d}(s), \theta_{t,\ell}) = \begin{cases} \left\lfloor \dfrac{2t - 5}{t - 2}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 1 \text{ and } t\ge 4, \\[10pt] \left\lfloor \dfrac{3t-10}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 2\text{ and } t\ge 6,\\[10pt] \left\lfloor \dfrac{(s + 1)t - (3s + 4)}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s \geq 3\text{ and } t\ge 7,\\[10pt] \left\lfloor \dfrac{2t - 7}{t - 3}d \right\rfloor + 1, & \text{if } s = 1 \text{ and } t\ge \max\{5,\ell+4\}, \end{cases} \] 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).