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A334613
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Number of distinct graceful labelings of bipartite graphs with n vertices and n-1 edges.
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2
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1, 1, 1, 2, 7, 28, 151, 907, 6467, 51151, 458478, 4455476, 48162974, 557022350, 7020022164, 94129389215, 1357181821028, 20661645476348
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OFFSET
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1,4
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COMMENTS
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Consider vertices numbered 1 to n. Add the edges 1--n, 2--n, and either 1--(n+1-k), 2--(n+2-k), ... or k--n for 3<=k<n. (Altogether (n-1)!/2 possibilities.) If the resulting graph has no odd cycles, this is a graceful labeling of a bipartite graph.
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LINKS
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EXAMPLE
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For example, the seven labelings for n=5 are:
1--5, 2--5, 1--3, 2--3;
1--5, 2--5, 1--3, 3--4;
1--5, 2--5, 1--3, 4--5;
1--5, 2--5, 2--4, 2--3;
1--5, 2--5, 2--4, 3--4;
1--5, 2--5, 3--5, 3--4;
1--5, 2--5, 3--5, 4--5.
The first of these is the four-cycle plus an isolated vertex; the other six are the trees in A337274.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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