An extension on neighbor sum distinguishing total coloring of graphs

Abstract

Let $f:V(G)\cup E(G)\to \{1,2,\dots ,k\}$ be a non-proper 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${}_{\sum }(G)$. The coloring is an extension of neighbor sum distinguishing non-proper total coloring. In this paper we conjecture that fgndi${}_{\sum }(G)\le 3$ for any connected graph $G$ of order at least three. We prove that the conjecture is true for (i) paths and cycles; (ii) 3-regular graphs and (iii) stars, complete graphs, trees, hypercubes, bipartite graphs and complete $r$-partite graphs. In particular, complete graphs can achieve the upper bound for the above conjecture.

arXiv admin note: text overlap with arXiv:2107.00424