A Brooks-type Theorem for the Bandwidth of Interval Graphs

Let G be an interval graph. The layout that arranges the intervals in order by right endpoint easily shows that the bandwidth of G is at most its maximum degree ∆. Hence, if G contains a clique of size ∆ + 1, then its bandwidth must be ∆. In this paper we prove a Brooks-type bound on the bandwidth of interval graphs. Namely, the bandwidth of an interval graph is at most ∆, with equality if and only if it does not contain a clique of size ∆ + 1. Furthermore, the stronger bound is tight even for interval graphs of clique number 2. Our proof utilizes the correspondence between the linear discrepancy of a partially ordered set and the bandwidth of its co-comparability graph. We also make progress toward a related question of Tanenbaum, Trenk, and Fishburn. They asked if a poset in which each point is incomparable to at most ∆ others has linear discrepancy at most b(3∆− 1)/2c. We show that this is true if the poset is disconnected.