Total interval numbers of complete r-partite graphs Original Research Article

A multiple-interval representation of a graph G is a mapping f which assigns to each vertex of G a union of intervals on the real line so that two distinct vertices u and v are adjacent if and only if f(u)∩f(v)≠∅. We study the total interval number of G, defined asI(G)=min∑v∈V #f(v): f is a multiple-interval representation of G,where #f(v) is the minimum number of intervals whose union is f(v). We give bounds on the total interval numbers of complete r-partite graphs. Exact values are also determined for several cases.