A Note on Linear Discrepancy and Bandwidth

Fishburn, Tanenbaum, and Trenk define the linear discrepancy \(\text{Id}(P)\) of a poset \( P = (V, <_P) \) as the minimum integer \( k \geq 0 \) for which there exists a bijection \( f : V \rightarrow \{1,2,\ldots,|V|\} \) such that \( u <_P v \) implies \( f(u) < f(v) \) and \( u ||_P v \) implies \( |f(u) – f(v)| \leq k \). In their work, they prove that the linear discrepancy of a poset equals the bandwidth of its cocomparability graph. Here we provide partial solutions to some problems formulated in their study about the linear discrepancy and the bandwidth of cocomparability graphs.