Degree complete graphs

Abstract

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘$\succcurlyeq$’ on the nonnegative integral vectors. It is clear that if $S=(s_1,s_2,\dots ,s_n)$ is an out-degree vector of an orientation of a graph G with vertices $1,2,\dots ,n$, then

(Π)$$S_G^r\succcurlyeq S\succcurlyeq S_G^l,\sum _{i=1}^n{s}_i=|E(G)|\text{and}0⩽s_i⩽d_G(i),i=1,2,\dots ,n\text{,}$$

where $S_G^r$ and $S_G^l$ are the maximum and minimum degree vectors with respect to ‘$\succcurlyeq$’, respectively. A graph G is called degree complete if each nonnegative integral vector satisfying the condition $(\mathrm{\Pi })$ is an out-degree vector of an orientation of G. By using flows in networks, the degree complete graphs are characterized by showing two simple forbidden configurations.