Conditions for sequences to be r-graphic

Abstract

In this paper two main results are obtained. One generalizes the well-known theorem of Erdös and Gallai [1] on the realizability of degree sequences by graphs without loops and multiple edges. It states that a nonincreasing sequence d = ( d 1, …, d n ) of nonnegative integers is r-graphic (that is, realizable by a loopless graph in which no two vertices are joined by more than r edges) if and only if (i) Σ n i=1 d i is even, and (ii) for every positive integer k ≤ n, ∑ i = 1 k d i ≤ r k ( k − 1 ) + ∑ i = k + 1 n min { r k , d i } The other result is a similar generalization of a theorem of Havel and Hakimi [5, 3].