The 1,2,3-Conjecture for powers of paths and powers of cycles
Resumo
A labelling of a graph G is a mapping π : S → L, where L ⊂ R and S = E(G) or S = V (G) ∪ E(G). If S = E(G), π is an L-edge-labelling and, if S = V (G) ∪ E(G), π is an L-total-labelling. For each v ∈ V (G), the colour of v under π is defined as cπ(v) = ∑uv∈E(G) π(uv) if π is an L-edge-labelling; and cπ(v) = π(v) + ∑uv∈E(G) π(uv) if π is an L-total-labelling. The pair (π, cπ) is a neighbour-distinguishing-L-edge (total)-labelling if π : S → L is an edge (total)-labelling and cπ(u) ≠ cπ(v), for every edge uv ∈ E(G). The 1,2,3-Conjecture states that every simple graph with no isolated edge has a neighbour-distinguishing-L-edge-labelling with L = {1, 2, 3}. In this work, we verify the 1,2,3-Conjecture for powers of paths and powers of cycles and we also show that powers of cycles have a neighbour-distinguishing-{a, b}-total-labelling, a, b ∈ R, a ≠ b.
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