Chartrand et al. (2004) have given an upper bound for the nearly antipodal chromatic number $a{c}^{\text{'}}\left(P_n\right)$ as $\left(\genfrac{}{}{0pt}{}{n-2}{2}\right)+2$ for $n\ge 9$ and have found the exact value of $a{c}^{\text{'}}\left(P_n\right)$ for $n=5,6,7,8$. Here we determine the exact values of $a{c}^{\text{'}}\left(P_n\right)$ for $n\ge 8$. They are $2p^2-6p+8$ for $n=2p$ and $2p^2-4p+6$ for $n=2p+1$. The exact value of the radio antipodal number $ac\left(P_n\right)$ for the path $P_n$ of order $n$ has been determined by Khennoufa and Togni in 2005 as $2p^2-2p+3$ for $n=2p+1$ and $2p^2-4p+5$ for $n=2p$. Although the value of $ac\left(P_n\right)$ determined there is correct, we found a mistake in the proof of the lower bound when $n=2p$ (Theorem $6$). However,...

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w\left(x\right)={\sum }_{y\in N_G\left(x\right)}\ell \left(y\right)$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover...

For a connected graph G of order n ≥ 3, let f: E(G) → ℤₙ be an edge labeling of G. The vertex labeling f’: V(G) → ℤₙ induced by f is defined as $f^{\text{'}}\left(u\right)={\sum }_{v\in N\left(u\right)}f\left(uv\right)$, where the sum is computed in ℤₙ. If f’ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) ≠ 0 for all e ∈ E(G) and in this case, G is a nowhere-zero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero...