Δ(G)=5的2-连通外平面图的邻点可区别全染色

Smarandachely邻点可区别全染色是指相邻点的色集合互不包含的邻点可区别全染色，是对邻点可区别全染色条件的进一步加强。本文研究了平面图的Smarandachely邻点可区别全染色，即根据2-连通外平面图的结构特点，利用分析法、数学归纳法，刻画了最大度为5的2-连通外平面图的Smarandachely邻点可区别全色数。证明了：如果$G$是一个$\Delta （G）=5$的2-连通外平面图，则$\chi_{\rm sat}（G）\leqslant 9$。

The Smrandachely adjacent vertex-distinguishing total coloring of 2-connected outer planar graphs with maximum degree 5

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 $\chi_{\rm sat}(G)\leqslant 9$.