On Eulerian Irregularities of Prisms, Grids and Powers of Cycles

Abstract

For a nontrivial connected graph $G$, an Eulerian walk in $G$ is a closed walk that contains every edge of $G$ at least once. An Eulerian walk is irregular if it encounters no two edges of $G$ the same number of times and the minimum length of an irregular Eulerian walk in $G$ is the Eulerian irregularity of $G$. In this work, we determine the Eulerian irregularities of all prisms, grids and powers of cycles.