The circular dimension of a graph

Abstract

A graph is a pair ( V, I), V being the vertices and I being the relation of adjacency on V. Given a graph G, then a collection of functions { f i } m n= 1 , each f i mapping each vertex of V into anarc on a fixed circle, is said to define an m-arc intersection model for G if for all x,y ϵ V, xly ⇔ (∨ i ⩽ m)( f i ( x)∩ f i ( y)≠Ø). The circular dimension of a graph G is defined as the smallest integer m such that G has an m-arc intersection model. In this paper we establish that the maximum circular dimension of any complete partite graph having n vertices is the largest integer p such that 2 p + p⩽ n+1.