Rank numbers of graphs that are combinations of paths and cycles

Abstract

A $k$-ranking of a graph $G$ is a function $f:V\left(G\right)\to \left\{1,2,\dots ,k\right\}$ such that if $f\left(u\right)=f\left(v\right)$, then every $u$-$v$ path contains a vertex $w$ such that $f\left(w\right)>f\left(u\right)$. The rank number of $G$, denoted ${\chi }_r\left(G\right)$, is the minimum $k$ such that a $k$-ranking exists for $G$. It is shown that given a graph $G$ and a positive integer $t$, the question of whether ${\chi }_r\left(G\right)\le t$ is NP-complete. However, the rank number of numerous families of graphs have been established. We study and establish rank numbers of some more families of graphs that are combinations of paths and cycles.