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 $\varphi$ such that, for every pair of adjacent edges $e$ and $f$, $|\varphi (e)|\ne |\varphi (f)|$. 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 $\Delta$, then $G$ admits a rich $(264\Delta -445)$-flow.

arXiv comment9 pages, 0 figures