Neighbor full sum distinguishing total coloring of Halin graphs

Let $ f: V(G)\cup E(G)\rightarrow \{1, 2, \dots, k\} $ be a total $ k $ -coloring of $ G $. Define a weight function on total coloring as

$ \phi(x) = f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y), $

where $ N(x) = \{y\in V(G)|xy\in E(G)\} $. If $ \phi(x)\neq \phi(y) $ for any edge $ xy\in 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 $ _{\sum}(G) $. Suppose that $ H = T\cup C $ is a Halin graph, where $ T $ and $ C $ are called the characteristic tree and the adjoint cycle, respectively. Let $ V_0\subseteq V(H)\setminus 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 $ _{\sum}(G)\leq 3 $ for any connected graph $ G $ of order at least three (Chang et al., 2022).