Extremal Values of the Interval Number of a Graph

Abstract

The interval number $i( G )$ of a simple graph G is the smallest number t such that to each vertex in G there can be assigned a collection of at most t finite closed intervals on the real line so that there is an edge between vertices v and w in G if and only if some interval for v intersects some interval for w. The well known interval graphs are precisely those graphs G with $i ( G )\leqq 1$. We prove here that for any graph G with maximum degree $d, i ( G )\leqq \lceil \frac{1}{2} ( d + 1 ) \rceil $. This bound is attained by every regular graph of degree d with no triangles, so is best possible. The degree bound is applied to show that $i ( G )\leqq \lceil \frac{1}{3}n \rceil $ for graphs on n vertices and $i ( G )\leqq \lfloor \sqrt{e} \rfloor $ for graphs with e edges.