Given a collection of n rooted trees with depth h, we give a necessary and sufficient condition for this collection to be the collection of h-depth universal covering neighborhoods at each vertex.
A short and constructive proof is available at @cite_1 . In fact, the Erdös-Gallai condition is not the only sufficient and necessary condition for an integer sequence to be graphical ; there are some other (equivalent) conditions, notably listed in @cite_7 . The corresponding realization problem for had also been solved quite early ; see the interesting note @cite_8 for a complete history and presentation of the many variants.
@cite_1 @cite_7 @cite_8
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Seven criteria for integer sequences being graphic are listed. Being graphic means that there is a simple graph with the given integer sequence as degree sequence. One of the criteria leads to a new and constructive proof of the well-known criterion of Erdos-Gallai. In 1930s Paul Erdős conjectured that for any positive integer C in any infinite ±1 sequence (x n ) there exists a subsequence x d , x 2d , x 3d ,…, x kd , for some positive integers k and d, such that ( i=1 ^k x_ id >C ). The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C = 1 a human proof of the conjecture exists; for C = 2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdős discrepancy conjecture for C = 2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C = 3. Abstract Given pairs ( a 1 , b 1 ) , … , ( a n , b n ) of nonnegative integers, the digraph realization problem for digraphs asks whether there is a simple digraph (no loops or multiple arcs) with vertices v 1 , … , v n such that each vertex v i ∈ V has indegree a i and outdegree b i . Fulkerson and Chen obtained a characterization analogous to the classical Erdős–Gallai characterization for graphs, but with the additional constraint that the pairs must be sorted in nonincreasing lexicographical order. We provide a more general characterization that avoids the additional sorting. The inequalities needed correspond to those k such that a k + 1 > a k . We prove a similar result when one loop is allowed at each vertex. Abstract Two conditions on a collection of simple orders - unimodality and straightness - are necessary but not jointly sufficient for unidimensional unfolding representations. From the analysis of these conditions, a polynomial time algorithm is derived for the testing of unidimensionality and for the construction of a representation when one exists.