For a connected graph G of order n ≥ 3, let f : E(G) → Z n be an edge labeling of G. The vertex labeling f ′ : V (G) → Z n induced by f is defined as f ′ (u) = v∈N (u) f (uv), where the sum is computed in Z n. 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 nowherezero modular edge-graceful graph. It is shown that a connected graph G of order n ≥ 3 is nowhere-zero modular edge-graceful if and only if n ≡ 2 (mod 4), G = K 3 and G is not a star of even order. For a connected graph G of order n ≥ 3, the smallest integer k ≥ n for which there exists an edge labeling f : E(G) → Z k − {0} such that the induced vertex labeling f ′ is one-to-one is referred to as the nowhere-zero modular edge-gracefulness of G and this number is determined for every connected graph of order at least 3.

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For a strong oriented graph D of order n and diameter d and an integerFor 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 = d D (u, v). The digraph D k is Hamiltonian-colored if D k contains a properly arc-colored Hamiltonian cycle. The smallest positive integer k for which D k is Hamiltonian-colored is the Hamiltonian coloring exponent hce(D) of D. For each integer n ≥ 3, the Hamiltonian coloring exponent of the directed cycle C n of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph D k such that hce(D k ) = k with the added property that every properly colored Hamiltonian cycle in the kth power of D k must use all k colors. It is shown for every positive integer p there exists a a connected graph G with two different strong orientations D and D ′ such that hce(D) − hce(D ′ ) ≥ p.

The squares of an m × n checkerboard are alternately colored black and red. It has been shown that for every pair m, n of positive integers, it is possible to place coins on some of the squares of the checkerboard (at most one coin per square) in such a way that for every two squares of the same color the numbers of coins on neighboring squares are of the same parity, while for every two squares of different colors the numbers of coins on neighboring squares are of opposite parity. All solutions to this problem have been what is referred to as trivial solutions, namely, for either black or red, no coins are placed on any square of that color. A nontrivial solution then requires at least one coin to be placed on a square of each color. For some pairs m, n of positive integers, however, nontrivial solutions do not exist. All pairs m, n of positive integers are determined for which there is a nontrivial solution. MSC2000: 05C15.