Multiplicity of Finite Graphs over the Real Line

Abstract

For topological spaces $X$ and $Y$, the multiplicity $m(X:Y)$ of $X$ over $Y$ is defined by M. Gromov and K. Taniyama independently. We show that the multiplicity $m(G:\mathbb{R}^1)$ of a finite graph $G$ over the real line $\mathbb{R}^1$ is equal to the cutwidth of $G$. We give a lower bound of $m(G:\mathbb{R}^1)$ and determine $m(G:\mathbb{R}^1)$ for an {\it $n$-constructed graph} $G$.