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A371514
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a(n) is the number of subsets of [floor(n/2)]* that are realizable by a graph G with n vertices.
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0
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OFFSET
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1,2
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COMMENTS
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A tree T with n vertices is bipartite. We write T=(X,Y) where the vertices of T in the first part are X, of order |X|=k, and the vertices in the second part are Y, of order |Y|=n-k. We arrange X and Y so that |X|<=|Y|. Then T has type (k,n-k), and we denote it by T_k. Let [floor(n/2)]*={(1,n-1),(2,n-2),...,(floor(n/2),ceiling(n/2))} be the set of all possible types for a graph with n vertices. For a connected graph G with n vertices, we define D(G)={(k,n-k)|G has a spanning tree T_k of type (k,n-k)}. For any subset S in [floor(n/2)]*, we say that S is realizable if there exists a graph G with n vertices and D(G)=S.
Realizable subsets will be featured in an upcoming article by Jayasooriya, McSorley, and Schuerger.
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LINKS
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EXAMPLE
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For n=6, a(6)=7. The set [floor(n/2)]* = {(1,5),(2,4),(3,3)}. So there are 8 subsets of [floor(n/2)]*. Out of those 8, the subset S = {(1,5),(3,3)} is not realizable. So a(6)=8-1=7.
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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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STATUS
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approved
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