Self-avoiding walks of specified lengths on rectangular grid graphs

Abstract

The investigation of self-avoiding walks on graphs has an extensive literature. We study the notion of wrong steps of self-avoiding walks on rectangular shape $$n\times m$$ n × m grids of square cells (Manhattan graphs) and examine some general and special cases. We determine the number of self-avoiding walks with one and with two wrong steps in general. We also establish some properties, like unimodality and sum of the rows of the Pascal-like triangles corresponding to the walks. We also present particular recurrence relations on the number of self-avoiding walks on the $$n\times 2$$ n × 2 grids with any specified number of wrong steps.