Degree bipartite Ramsey numbers

Let $H\xrightarrow{s} G$ denote that any edge-coloring of $H$ by $s$ colors contains a monochromatic $G$. The degree Ramsey number $r_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\}$, and the degree bipartite Ramsey number $br_{\Delta}(G;s)$ is defined to be $\min\{\Delta(H):H\xrightarrow{s} G\; \mbox{and} \;\chi(H)=2\}$. In this note, we show that $r_{\Delta}(K_{m,n};s)$ is linear on $n$ with $m$ fixed. We also determine $br_{\Delta}(G;s)$ where $G$ are trees, including stars and paths, and complete bipartite graphs.