On $S$-packing total colorings

In this paper, we generalize the concept of packing total coloring by introducing a new concept called the $S$-packing total coloring. For a graph $G$ and a non-decreasing sequence $S=(a_1,a_2,\ldots)$ of positive integers, an $S$-packing total coloring of $G$ is a mapping $c: V(G)\cup E(G)\rightarrow \{1,2,\ldots\}$ such that for any two distinct elements $A,B\in V(G)\cup E(G)$ with $c(A)=c(B)=i$, the distance between $A$ and $B$ is at least $a_i+1$. The smallest integer $k$ such that $G$ admits an $S$-packing total coloring using $k$ colors is called the $S$-packing total chromatic number of $G$, denoted by $χ_S^{''}(G)$. For any sequence $S$, we establish general lower and upper bounds for $χ_S^{''}(G)$, and characterize all graphs $G$ with $χ_S^{''}(G)\in\{1,2,3\}$. Furthermore, we investigate $S$-packing total chromatic numbers of complete bipartite graphs, as well as infinite and finite paths and cycles.