The local crossing number is defined as the least nonnegative integer such that the graph has a -planar drawing. In other words, it is the smallest possible
number of times that a single edge in a graph is crossed over all possible graph
drawings. Guy et al. (1968) attribute the definition to unpublished work of
Ringel.

The local crossing number of a graph is called the cross-index by Thomassen (1988) and sometimes also the crossing parameter (Schaefer 2013), but Schaefer (2013) strongly encourages the use of "local crossing number." However, the term "planarity" might be more more descriptive and more concise.

Schaefer (2014) and Ábrego and Fernández-Merchant (2017) denote the local crossing number of a graph as .

Graphs with local crossing number 0 are equivalent to planar graphs. In general, a k-planar graph
can have local crossing number 0, 1, ..., or .

Ábrego, B. M. and Fernández-Merchant, S. "The Rectilinear Local Crossing Number of ." J. Combin. Th. Ser. A151, 131-145,
2017.de Klerk, E.; Pasechnik, D. V.; and Schrijver, A. "Reduction
of Symmetric Semidefinite Programs Using the Regular -Representation." Math. Program.109, 613-624,
2007.Guy, R. .K; Jenkyns, T.; and Schaer, J. "The Toroidal
Crossing Number of the Complete Graph." J. Combin. Th.4, 376-390,
1968.Kainen, P. C. "Thickness and Coarseness of Graphs."
Abh. Math. Semin. Univ. Hamburg39, 88-95, 1973.Ringel,
G. "Ein Sechsfarbenproblem auf der Kugel." Abh. Math. Sem. Univ. Hamburg29,
107-117, 1965.Schaefer, M. "The Graph Crossing Number and Its Variants:
A Survey." Electron. J. Combin., DS21, pp. 43-45, May 15, 2013.Thomassen,
C. "Rectilinear Drawings of Graphs." J. Graph Th.12, 335-341,
1988.