A Note on the Adjacent Vertex Distinguishing Total Coloring of Special Planar Graphs

An adjacent vertex distinguishing total coloring of a graph \(G\) is a proper total \(k\)-coloring of \(G\) such that any two adjacent vertices have different color sets, where the color set of a vertex \(v\) contains the color of \(v\) and the colors of its incident edges. Let \(\chi_{a}^{”}(G)\) denote the smallest value \(k\) in such a coloring of \(G\). In this paper, by using the Combinatorial Nullstellensatz and the discharging method, we prove that if a planar graph \(G\) with maximum degree \(\Delta \geq 9\) contains no \(5\)-cycles with more than one chord, then \(\chi_{a}^{”}(G) \leq \Delta + 3\).