Cyclic-order graphs and Zarankiewicz's crossing-number conjecture

Zarankiewicz's conjecture, that the crossing number of the completebipartite graph K,,,, is [$ rnllfr (m - 1)Jl; nj[$ (n - 1)j, was proved by Kleitman when min(rn, n) s 6, but was unsettled in all other cases. The cyclic-order graph CO, arises naturally in the study of this conjecture; it is a vertex-transitive harmonic diametrical (even) graph. In this paper the properties of cyclic-order graphs are investigated and used as the basis for computer programs that have verified Zarankiewicz's conjecture for K7,7 and K7,9; thus the smallest unsettled cases are now K7,11 and K9,9. 0 1993 John Wiley & Sons, Inc.

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