Total colorings of planar graphs without adjacent triangles (Q998509)

Let \(G=\left(V(G), E(G)\right)\) be a graph with maximum degree \(\Delta\). A total \(k\)-coloring of \(G\) is a coloring of \(V(G) \cup E(G)\) using \(k\) colors such that no two adjacent or incident elements receive the same color. The total chromatic number \(\chi''(G)\) is the smallest integer \(k\) such that \(G\) has a total \(k\)-coloring. The well-known Total Coloring Conjecture asserts that \(\Delta+1 \leq \chi''(G) \leq \Delta+2\). A partial result is established in this paper as follows. The Total Coloring Conjecture is true for a planar graph \(G\) without adjacent 3-cycles. Furthermore, \(\chi''(G)=\Delta+1\) if \(\Delta \geq 9\).