AROUND THE ERDÖS–GALLAI CRITERION

Abstract

By an (integer) partition we mean a non-increasing sequence \(\lambda=(\lambda_1, \lambda_2, \dots)\) of non-negative integers that contains a finite number of non-zero components. A partition \(\lambda\) is said to be graphic if there exists a graph \(G\) such that \(\lambda = \mathrm{dpt}\,G\), where we denote by \(\mathrm{dpt}\,G\) the degree partition of \(G\) composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper.

Keywords

• integer partition, threshold graph, bipartite graph, bipartite-threshold graph, ferrers diagram