Abstract

This paper investigates crossing numbers of join products involving all non-isomorphic graphs of order five and the graph families Dn, Pn, and Cn. By systematically examining all 34 graphs of order five, we establish a unified classification of 102 crossing number values together with the corresponding references and principal proof techniques. The collected results consolidate previously scattered contributions, clarify several inconsistencies in the literature, and identify the unique remaining unresolved case. For the graph H57, we verify the conjectured crossing numbers computationally for all feasible values n≤5, establish general upper bounds, and prove that the corresponding results for joins with paths and cycles follow directly from the discrete graph case. Consequently, determining the exact crossing numbers of H57 is reduced to establishing the corresponding lower bounds for joins with discrete graphs. The resulting classification reveals that the crossing number values are governed by only a small number of recurring formula types, suggesting that the crossing behaviour of join products involving graphs of order five is determined primarily by a limited collection of fundamental structural patterns.

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