Geodesic
The Coarseness of the Complete Bipartite Graph
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1086-1096

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The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar graphs whose union is G. The coarseness of the complete graph has been investigated elsewhere (1; 2). We consider the coarseness of the complete bipartite, or 2-coloured, graph, Km,n , consisting of sets of mand nvertices, each member of one set being joined by an edge to each member of the other. No members of one set are joined to each other.Our results are summarized in the following theorem, where square brackets denote “integer part”.THEOREM. If m= 3p + d, 0 ≦ d≦ 2, and n = 3q + e, 0 ≦ e ≦ 2, then for d = 0 or 1 and e = 0 or 1, 1 
Beineke, Lowell W.; Guy, Richard K. The Coarseness of the Complete Bipartite Graph. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 1086-1096. doi: 10.4153/CJM-1969-121-3
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[1] 1. Guy, R. K., On a coarseness conjecture of Erdôs, J. Combinatorial Theory 3 (1967), 38–42. Google Scholar

[2] 2. Guy, R. K. and Beineke, L. W., The coarseness of the complete graph, Can. J. Math. 20 (1968), 888–894. Google Scholar

[3] 3. Kuratowski, K., Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283. Google Scholar

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