@article{Andreae1989TheTI,
title={The total interval number of a graph},
author={Thomas Andreae and Martin Aigner},
journal={J. Comb. Theory, Ser. B},
year={1989},
volume={46},
pages={7-21}
}

A representation f of a graph G is a mapping f which assigns to each vertex of G a non-empty collection of intervals on the real line so that two distinct vertices x and y are adjacent if and only if there are intervals I~f(x) and J~f(y) with InJ# a. We study the total interval number of G, defined as I(G) = minCL, VtGj If(x)l:fis a representation of G}. Let n be the number of vertices of G. Our main results on I(G) are: (a) If G is a tree then I(G)<L5n/4-3/4J for n > 3. (b) If G is a triangle… CONTINUE READING