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

Abstract: A rich $k$-flow is a nowhere-zero $k$-flow $φ$ such that, for every pair of adjacent edges $e$ and $f$, $|φ(e)| \neq |φ(f)|$. 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 $Δ$, then $G$ admits a rich $(264Δ- 445)$-flow.