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, …, k}, such that |w(x) − w(y)| ≤ 1, for every adjacent x, y ∈ V. We find the explicit generating functions of the number of staircase graph words for the grid graph, the rectangle-triangular graph and the king’s graph, all of size 2 × n.