Neighbor full sum distinguishing total coloring of Halin graphs

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

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

where $N(x)=\{y\in V(G)|xy\in E(G)\}$. If $\varphi (x)\ne \varphi (y)$ 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 ${}_{\sum }(G)\le 3$ for any connected graph $G$ of order at least three (Chang et al., 2022).