On graceful coloring of generalized Petersen graphs

Abstract

The graphs used in this paper are simple, connected and finite graph. A graceful [Formula: see text]-coloring of a graph [Formula: see text] is a proper vertex coloring [Formula: see text], where [Formula: see text] which induces a proper edge coloring [Formula: see text] that defined by [Formula: see text]. Proper vertex coloring [Formula: see text] of a graph [Formula: see text] is a graceful coloring if [Formula: see text] is a graceful m-coloring for [Formula: see text]. The minimum vertex coloring, that can apply such that a graph [Formula: see text] can be colored with graceful coloring, is called a graceful chromatic number and it is denoted by [Formula: see text]. In this paper, we will investigate the graceful chromatic number of generalized Petersen graph [Formula: see text] for [Formula: see text].