The Total Interval Number of a Graph

Abstract

An interval representation (or simply representation) R of a graph G is a collection of finite sets $\{R(\nu):\nu \in V(G)\}$ of closed bounded intervals so that $u \leftrightarrow \nu$ if and only if there exist $\theta\sb{u} \in R(u), \theta\sb{\nu} \in R(\nu)$ with $\theta\sb{u} \cap \theta\sb{\nu} \not= \emptyset$. The size of a representation is the number of intervals in the entire collection.

The total interval number of G is the size of the smallest representation of G and is denoted I(G). This thesis studies I by proving best possible upper bounds for several classes of graphs. For some classes, the bounds are in terms of n, the number of vertices and for some classes, the bounds are in terms of m, the number of edges. The main result is that for planar graphs, $I(G) \leq 2n(G) - 3$.