An upper bound for the crossing number of bubble-sort graph Bn

The crossing number of a graph G is the minimum number of pairwise intersections of edges in a drawing of G. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt’o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory 59, 145-161 (2008)], we give an upper bound of the crossing number of n-dimensional bubble-sort graph Bn.

💡 Research Summary

The paper addresses the long‑standing problem of estimating the crossing number cr(G) of the n‑dimensional bubble‑sort graph Bₙ, a Cayley graph of the symmetric group Sₙ generated by the adjacent transpositions (i,i+1). Bₙ has |V| = n! vertices and degree n − 1, and it appears in parallel sorting networks, interconnection topologies, and combinatorial optimization. The crossing number, defined as the minimum possible number of pairwise edge intersections in any planar drawing of G, directly influences VLSI layout cost, wiring length, and visual readability.

The authors first observe that each vertex (a permutation) can be classified by its inversion number, i.e., the number of pairs (i,j) with i < j but σ(i) > σ(j). This yields a natural layering of Bₙ into n + 1 levels L₀,…,Lₙ, where L_k consists of all permutations with exactly k inversions. Crucially, an adjacent transposition changes the inversion number by exactly ±1, so edges of Bₙ run only between consecutive layers. This “inversion‑level” decomposition turns Bₙ into a layered graph reminiscent of the permutohedron.

Using this structure, the authors construct an explicit drawing. Vertices of each layer L_k are placed uniformly on a circle (or equally spaced on a horizontal line). All edges between L_k and L_{k+1} are drawn as identical convex arcs (for example, upward‑facing Bézier curves). Because edges never stay within a single layer, intra‑layer crossings are eliminated. Crossings can only occur between edges belonging to different layer pairs (i,j) with i < j. For a fixed pair (i,j) the number of crossing pairs is exactly

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