Abstract:

The adjacent vertex distinguishable total coloring is the total coloring with different sets of adjacent vertices, while the Smarandachely adjacent vertex distinguishing total coloring is the adjacent vertex distinguishing total coloring which does not contain each other in the color set of adjacent vertices. It is a further enhancement of the condition of adjacent vertex-distinguishing total coloring. We call the minimum number of colors used for a graph to satisfy Smarandachely's adjacent vertex-distinguishing total coloring as its Smarandachely adjacent vertex-distinguishing total chromatic number. In this paper, the Smarandachely adjacent vertex distinguishing total chromatic number of 2-connected outer plane graphs with maximum degree of 5 is studied by means of analytical method and mathematical induction. It is proved that if $G$ is a 2-connected outer planar graph with $\Delta(G)=5 $, then $\chi_{\rm sat}(G)\leqslant 9$.

Key words: outerplanar graph, Smarandachely adjacent-vertex distinguishing total coloring, Smarandachely adjacent-vertex distinguishing total chromatic number