Bounded-Degree Graphs have Arbitrarily Large Queue-Number

We consider graphs possibly with loops but with no parallel edges. A graph without loops is simple. Let G be a graph with vertex set V (G) and edge set E(G). If S ⊆ E(G) then G[S] denotes the spanning subgraph of G with edge set S. We say G is ordered if V (G) = {1, 2, . . . , |V (G)|}. Let G be an ordered graph. Let `(e) and r(e) denote the endpoints of each edge e ∈ E(G) such that `(e) ≤ r(e). Two edges e and f are nested and f is nested inside e if `(e) < `(f) and r(f) < r(e). An ordered graph is a queue if no two edges are nested. Observe that the left and right endpoints of the edges in a queue are in first-in-first-out order—hence the name ‘queue’. An ordered graph G is a k-queue if there is a partition {E1, E2, . . . , Ek} of E(G) such that each G[Ei] is a queue. Let G be an (unordered) graph. A k-queue layout of G is a k-queue that is isomorphic to G. The queue-number of G is the minimum integer k such that G has a k-queue layout. Queue layouts and queuenumber were introduced by Heath et al. [15, 16] in 1992, and have applications in sorting permutations [12, 17, 23, 25, 29], parallel process scheduling [3], matrix computations [24], and graph drawing [4, 6]. Other aspects of queue layouts have been studied in [7, 8, 10, 14, 26, 27, 30].