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The coarseness of a graph is the maximum number of edge-disjoint nonplanar subgraphs contained in a given graph . The coarseness of a planar graph is therefore .
The coarseness of a graph is the sum of the coarsenesses of its blocks (Beineke and Chartrand 1968).
The coarseness of the complete graph is known for most values of except , divisible by 3 and greater than or equal to 18, and of the form . For all of these, the values are known to within 1 (Guy and Beineke 1968; Harary 1994, pp. 121-122).
The coarseness of the complete bipartite graph is known for values of satisfying certain conditions (Beineke and Guy 1969; Harary 1994, pp. 121-122).
Beineke, L. W. and Chartrand, G. "The Coarseness of a Graph." Compos. Math. 19, 290-298, 1968.
Beineke, L. W. and Guy, R. K. "The Coarseness of the Complete Bipartite Graph." Canad. J. Math. 21, 1086-1096, 1969.
Guy, R. and Beineke, L. "'THe Coarseness of the Complete Graph." Canad. J. Math. 20, 888-894, 1968.
Harary, F. "Covering and Packing in Graphs, I." Ann. New York Acad. Sci. 175, 198-205, 1970.
Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 121-122, 1994.
Harary, F. and Palmer, E. M. Graphical Enumeration. New York: Academic Press, p. 225, 1973.
Harary, F. and Palmer, E. M. "A Survey of Graph Enumeration Problems." In A Survey of Combinatorial Theory (Ed. J. N. Srivastava). Amsterdam: North-Holland, pp. 259-275, 1973.
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