On the Vertex-Distinguishing Proper Total Colorings of Complete \(p\)-Partite Graphs with Equipotent Parts

Abstract

A proper \(k\)-total coloring of a simple graph \(G\) is called \(k\)-vertex-distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The minimum number of colors required for a vertex-distinguishing proper total coloring of \(G\), denoted by \(\chi_{vt}(G)\), is called the vertex-distinguishing proper total chromatic number. For \(p\) even, \(p \geq 4\) and \(q \geq 3\), we will obtain vertex-distinguishing proper total chromatic numbers of complete \(p\)-partite graphs with each part of cardinality \(q\).