Abstract
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 as I(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.
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