Adjacent vertex distinguishing total coloring in split graphs

Abstract

An adjacent vertex distinguishing (AVD-) total coloring of a graph G is a total coloring such that any two adjacent vertices u and v have distinct sets of colors, that is, C(u)≠C(v), where C(v) is the set of colors of the edges incident to v and the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of a graph G, χa″(G) is the minimum integer k such that there exists an AVD-total coloring of G using k colors. It is known that χa″(G)≥Δ+1, where Δ is the maximum degree of the graph. The AVD-total coloring conjecture states that for any graph G, χa″(G)≤Δ+3. In this paper, we study AVD-total coloring in split graphs. We verify the AVD-total coloring conjecture for split graphs and classify certain classes of split graphs according to their AVD-total chromatic number.