Regular subgraphs of almost regular graphs

Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k > 2q 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q. r = q (mod 2)). It is also proved that every simple graph with maximal degree A > 2q 2 and average degree d > ((2q 2)/(2q l))(A + 1), where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r = q (mod 2)). These results follow from Chevalley’s and Olson’s theorems on congruences.