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 $\min \{ \Delta(H): H G \}$, and the bipartite $br_{\Delta}(G;s)$ \textrm{ } \chi(H) = 2 \}$. In this note, we show $r_{\Delta}(K_{m,n};s)$ linear on $n$ with fixed $m$. We also evaluate for paths other trees.