A tree with n vertices is called graceful, if its vertices can be labelled with integers 1, 2,....n such that the absolute value of the difference of the labels of adjacent vertices are all different. Which of the following trees are graceful ? Codes :
(a), (b) and (c)
A tree on n vertices has n - 1 edges, so a graceful labelling must produce the edge differences 1 to n - 1 exactly once.
Tree (a): label \(v_{1}, v_{2}, v_{3}, v_{4}\) as 1, 4, 2, 3; the differences are 3, 2, 1.
Tree (b): label \(v_{1}, v_{2}, v_{3}, v_{4}, v_{5}\) as 5, 1, 4, 3, 2; the edges \(v_{1}v_{2}, v_{2}v_{3}, v_{2}v_{4}, v_{4}v_{5}\) give 4, 3, 2, 1.
Tree (c): label \(v_{1}, \ldots, v_{6}\) as 2, 6, 1, 3, 4, 5; the edges \(v_{1}v_{2}, v_{2}v_{3}, v_{3}v_{4}, v_{3}v_{5}, v_{5}v_{6}\) give 4, 5, 2, 3, 1.
Check: all differences are distinct in each tree, so all three trees are graceful.
Hence, (a), (b) and (c).
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