@article{Cheng2022, 
author = {Yinwan Cheng and Chao Yang and Bing Yao and Yaqin Luo},
title = {Neighbor full sum distinguishing total coloring of Halin graphs},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {4},
pages = {6959-6970},
keywords = {total coloring, neighbor full sum distinguishing total coloring, neighbor full sum distinguishing total chromatic number, Halin graphs},
url = {https://www.sciopen.com/article/10.3934/math.2022386},
doi = {10.3934/math.2022386},
abstract = {Let    f  :  V  (  G  )  ∪  E  (  G  )  →  {  1  ,  2  ,  …  ,  k  } be a total    k -coloring of    G. Define a weight function on total coloring as     ϕ  (  x  )  =  f  (  x  )  +      ∑          e      ∋      x        f  (  e  )  +      ∑          y      ∈      N      (      x      )        f  (  y  )  ,where    N  (  x  )  =  {  y  ∈  V  (  G  )      |    x  y  ∈  E  (  G  )  }. If    ϕ  (  x  )  ≠  ϕ  (  y  ) for any edge    x  y  ∈  E  (  G  ), then    f is called a neighbor full sum distinguishing total    k -coloring of    G. The smallest value    k for which    G has such a coloring is called the neighbor full sum distinguishing total chromatic number of    G and denoted by fgndi                  ∑        (  G  ). Suppose that    H  =  T  ∪  C is a Halin graph, where    T and    C are called the characteristic tree and the adjoint cycle, respectively. Let        V    0    ⊆  V  (  H  )  ∖  V  (  C  ) and each vertex in        V    0   is adjacent to some vertices on    C. In this paper, we prove that the neighbor full sum distinguishing total chromatic number of two types of Halin graphs are not more than three: (i) 3-regular Halin graphs and (ii) every vertex of        V    0   of a Halin graph with degree at least 4. The above results support a conjecture that fgndi                  ∑        (  G  )  ≤  3 for any connected graph    G of order at least three (Chang et al., 2022).}
}