Lin SUN, De-rong SUN, Xin LI, Guang-long YU
Given a simple graph $G=(V, E)$ and its (proper) total coloring $\phi$ with elements of the set $\{1, 2,\cdots, k\}$, let $w_{\phi}(v)$ denote the sum of the color of $v$ and the colors of all edges incident with $v$. If for each edge $uv\in E$, $w_{\phi}(u)\neq w_{\phi}(v)$, we call $\phi$ a neighbor sum distinguishing total coloring of $G$. Let $L=\{L_x\, |\, x\in V\cup E\}$ be a set of lists of real numbers, each of size $k$. The neighbor sum distinguishing total choosability of $G$ is the smallest $k$ for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from $L_x$ for each $x\in V\cup E$, and we denote it by ${\rm ch}''_{\sum}(G)$. The known results of neighbor sum distinguishing total choosability are mainly about planar graphs. In this paper, we focus on $1$-planar graphs. A graph is $1$-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. We prove that ${\rm ch}''_{\sum}(G)\leq \Delta+4$ for any $1$-planar graph $G$ with $\Delta\geq 15$, where $\Delta$ is the maximum degree of $G$.
