A Linear Bound on the Rich Flow Number for Graphs with a Given Maximum Degree

A rich $k$-flow is a nowhere-zero $k$-flow $\phi$ such that, for every pair of adjacent edges $e$ and $f$, $\mathrm{\mid }\phi (e)\mathrm{\mid }\mathrm{\ne }\mathrm{\mid }\phi (f)\mathrm{\mid }$. A graph is rich flow admissible if it admits a rich $k$-flow for some integer $k$. In this paper, we prove that if $G$ is a rich flow admissible graph with maximum degree $\mathrm{\Delta }$, then $G$ admits a rich $(264\mathrm{\Delta }-445)$-flow.