Abstract
Coloring is a fundamental problem in scientific computation and engineering design. In recent years, a variety of colorings frequently appeared and solved many problems in production. For example, adjacent-vertex-distinguishing proper total coloring, adjacent-vertex-distinguishing proper edge coloring, smarandachely-adjacent-vertex-distinguishing proper edge coloring. It is an important also difficult problem to discuss the coloring numbers of a given graph class. And we focus on the adjacent-vertex-distinguishing proper total coloring numbers in this paper. We study the adjacent-vertex-distinguishing proper total coloring numbers of joint graphs
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