On existence of hypergraphs with prescribed edge degree profile

The degree profile of an edge e of a finite hypergraph H is the map assigning to a positive integer i the number of vertices of degree i incident with e. The edge degree profile of H is the map describing for any possible degree profile ~ the number of edges in H with degree profile ~. A necessary and sufficient condition for existence of hypergraphs of prescribed edge degree profile is found. A similar result concerning hypergraphs without multiple edges is also presented. One of basic informations about a simple graph G (unordered, finite, without loops and multiple edges) is contained in the sequence (vl(G) . . . . . vA(G)) where vi(G) is the number of i-valent vertices of G, i = 0 . . . . . A = A (G). A natural problem to characterize all finite sequences (Vo . . . . . va) of non-negative integers with va > 0 such that there is a graph G with A(G) = A and vi(G) = vi, i = 0 . . . . ,A, has been first solved independently by Havel [8] and Hakimi [6]. It is well known that a graph corresponding to a sequence (Vo . . . . ,VA), if any, need not be unique. A more complex information on the structure of G is yielded by the symmetric matrix (eij(G)) of order A(G) having in ith row and jth column the number e~(G) of edges of G joining an i-valent vertex to a j-valent vertex. (All v~(G) are derivable from this matrix.) Hakimi and Schmeichel [7] raised a problem analogous to that above, namely: Given a symmetric matrix g = (e 0 of order A with Z~= 1 e~a > 0, does there exist a graph G with A(G) = A such that eij(G) = e i j , i,j = 1 . . . . . A? Note that without loss of generality G can be assumed to be without isolated vertices they do not influence eij(G). A necessary and sufficient condition was found by Patrinos and Hakimi [9] it can be reformulated as the conjunction of two following * Corresponding author. 0012-365X/95/$09.50 © 1995--Elsevier Science B.V. All rights reserved SSDI 0012-365X(94)001 58-8 110 M. Horhftk et al. / Discrete Mathematics 146 (1995) 109-121