Online Ramsey numbers of the claw versus cycles

The online Ramsey number $\tilde{r}(G,H)$ is defined via a Builder–Painter game on an empty graph with countably many vertices. In each round, Builder reveals an edge, which Painter immediately colors either red or blue. Builder wins once a red copy of $G$ or a blue copy of $H$ appears, and $\tilde{r}(G,H)$ is the minimum number of edges Builder must reveal to force a win. For a long cycle $C_{\mathrm{\ell }}$, the online Ramsey numbers $\tilde{r}(G,C_{\mathrm{\ell }})$ are known only for a few specific choices of $G$. In particular, exact values were determined for $G=P_3$ by Cyman, Dzido, Lapinskas, and Lo (Electron. J. Combin., 2015), while asymptotically tight results were obtained when $G$ is an even cycle by Adamski, Bednarska-Bzdȩga, and Blažej (SIAM J. Discrete Math., 2024). In this paper, we consider the case where $G$ is the claw $K_{1,3}$ and determine the exact value of $\tilde{r}(K_{1,3},C_{\mathrm{\ell }})$. We show that $$\tilde{r}(K_{1,3},C_{\mathrm{\ell }})=\lfloor \frac{3(\mathrm{\ell }+1)}{2}\rfloor \text{for all }\mathrm{\ell }\ge 13.$$