Disjoint subsets of integers having a constant sum
Discrete Mathematics
A note on ascending subgraph decompositions of complete multipartite graphs
Discrete Mathematics
Ascending subgraph decomposition of regular graphs
Discrete Mathematics
Combinatorics, Probability and Computing
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A sequence m"1=m"2=...=m"k of k positive integers isn-realizable if there is a partition X"1,X"2,...,X"k of the integer interval [1,n] such that the sum of the elements in X"i is m"i for each i=1,2,...,k. We consider the modular version of the problem and, by using the polynomial method by Alon (1999) [2], we prove that all sequences in Z/pZ of length k@?(p-1)/2 are realizable for any prime p=3. The bound on k is best possible. An extension of this result is applied to give two results of p-realizable sequences in the integers. The first one is an extension, for n a prime, of the best known sufficient condition for n-realizability. The second one shows that, for n=(4k)^3, an n-feasible sequence of length k isn-realizable if and only if it does not contain forbidden subsequences of elements smaller than n, a natural obstruction forn-realizability.