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Let $G$ be a simple graph. For a general edge coloring of a graph $G$ (i.e., not necessarily a proper edge coloring) and a vertex $v$ of $G$, denote by $S\left(v\right)$ the set (not a multiset) of colors used to color the edges incident to $v$. For a general edge coloring $f$ of a graph $G$, if $S\left(u\right)\ne S\left(v\right)$ for any two different vertices $u$ and $v$ of $G$, then we say that $f$ is a point-distinguishing general edge coloring of $G$. The minimum number of colors required for a point-distinguishing general edge coloring of $G$, denoted by ${\chi }_0\left(G\right)$, is called...

Let G be a simple graph. An IE-total coloring f of G is a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C(u) be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C(u) 6= C(v) for any two different vertices u and v of G, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring...