On smarandachely adjacent vertex total coloring of subcubic graphs

Inspired by the observation that adjacent vertices need possess their own characteristics in terms of total coloring, we study the smarandachely adjacent vertex total coloring (abbreviated as SAVTC) of a graph G , which is a proper total coloring of G such that for every vertex u and its every neighbor v , the color-set of u contains a color not in the color-set of v , where the color-set of a vertex is the set of colors appearing at the vertex or its incident edges. The minimum number of colors required for an SAVTC is denoted by χ s a t ( G ) . Compared with total coloring, SAVTC would be more likely to be developed for potential applications in practice. For any graph G , it is clear that χ s a t ( G ) ≥ Δ ( G ) + 2 , where Δ ( G ) is the maximum degree of G . We, in this work, analyze this parameter for general subcubic graphs. We prove that χ s a t ( G ) ≤ 6 for every subcubic graph G . Especially, if G is an outerplanar or claw-free subcubic graph, then χ s a t ( G ) = 5 .