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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...

For a strong oriented graph D of order n and diameter d and an integer k with 1 ≤ k ≤ d, the kth power $D^k$ of D is that digraph having vertex set V(D) with the property that (u, v) is an arc of $D^k$ if the directed distance ${}^{\to }{d}_D(u,v)$ from u to v in D is at most k. For every strong digraph D of order n ≥ 2 and every integer k ≥ ⌈n/2⌉, the digraph $D^k$ is Hamiltonian and the lower bound ⌈n/2⌉ is sharp. The digraph $D^k$ is distance-colored if each arc (u, v) of $D^k$ is assigned the color i where $i{=}^{\to }{d}_D(u,v)$. The digraph $D^k$ is Hamiltonian-colored...