Dense Arbitrarily Partitionable Graphs

Abstract

A graph G of order n is called arbitrarily partitionable (AP for short) if, for every sequence (n1, . . . , nk) of positive integers with n1 + ⋯ + nk = n, there exists a partition (V1, . . . , Vk) of the vertex set V (G) such that Vi induces a connected subgraph of order ni for i = 1, . . . , k. In this paper we show that every connected graph G of order n ≥ 22 and with ‖G‖ > (n−42)+12$||G||\; > \;\left( {\matrix{{n - 4} \cr 2 \cr } } \right) + 12$ edges is AP or belongs to few classes of exceptional graphs.

Keywords

• arbitrarily partitionable graph
• erdös-gallai condition
• traceable graph
• perfect matching
• 05c38
• 05c70