The Local Distinguishing Number of Polygonal Prisms

Abstract

A polygonal prism with 2n vertices, denoted by PPn, is a product graph of a Cn and a K2. From the outward appearance, a polygonal prism PPn consists of two base faces and n lateral faces. Each lateral face is an induced square, or a 4-cycle, of PPn. To distinguish all of the lateral faces of PPn, we color every vertex in the graph. Two lateral faces are distinguishable in colors if one face cannot be obtained from the other by rotating and flipping. As a result, the local distinguishing number problem with respect to a polygonal prism, or LDPP for short, is to determine the minimum number of colors required to distinguish each lateral face of the polygonal prism. The LDPP is a variation of graph coloring problem. In this paper, we propose a coloring algorithm and bound the local distinguishing number of a polygonal prism.