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Nowhere-zero modular edge-graceful graphs

100%

Jones R., Zhang P.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 3

487-505

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 $f'(u) = ∑_{v ∈ N(u)} f(uv)$, 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 modular edge-graceful if and only if n ≢ 2 mod 4, G ≠ K₃ 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) → ℤₖ - {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.

Hamiltonian-colored powers of strong digraphs

81%

Johns G., Jones R., Kolasinski K., Zhang P.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 4

705-724

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 $^{→}d_D(u,v)$ 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 $i = ^{→}d_D(u,v)$. 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 $^{→}Cₙ$ of order n is determined whenever this number exists. It is shown for each integer k ≥ 2 that there exists a strong oriented graph Dₖ such that hce(Dₖ) = k with the added property that every properly colored Hamiltonian cycle in the kth power of Dₖ 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.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Jones R.

2 Zhang P.

1 Johns G.

1 Kolasinski K.

2 2012

Wyniki wyszukiwania

Nowhere-zero modular edge-graceful graphs

Hamiltonian-colored powers of strong digraphs