AVDTC numbers of generalized Halin graphs with maximum degree at least 6

In a paper by Zhang and Chen et al.(see [11]), a conjecture was made concerning the minimum number of colors _{χ at} (G) required in a proper total-coloring of G so that any two adjacent vertices have different color sets, where the color set of a vertex ν is the set composed of the color of ν and the colors incident to ν. We find the exact values of _{χ at} (G) and thus verify the conjecture when G is a Generalized Halin graph with maximum degree at least 6. A generalized Halin graph is a 2-connected plane graph G such that removing all the edges of the boundary of the exterior face of G (the degrees of the vertices in the boundary of exterior face of G are all three) gives a tree. Supported by the National Natural Science Foundation of China (No.10771091) and the Science and Research Project of the Education Department of Gansu Province (0501-02) and NWNU-KJCXGC-3-18.