Staircase graph words

Generalizing the notion of staircase words, introduced by Knopfmacher et.\ al, we define staircase graph words. These are functions $w$ from the vertex set $V$ of a graph into the set $\{1,2,\dots ,k\}$, such that $\mathrm{\mid }w(x)-w(y)\mathrm{\mid }\le 1$, for every adjacent $x,y\in V$. We find the explicit generating functions for the number of staircase graph words for the grid graph, the rectangle-triangular graph and the king’s graph, all of size $2\times n$.