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The Coarseness of the Complete Bipartite Graph

Published online by Cambridge University Press:  20 November 2018

Lowell W. Beineke  and
Richard K. Guy
Lowell W. Beineke
Affiliation:
Purdue University, Fort Wayne, Indiana
Richard K. Guy
Affiliation:
University of Calgary, Calgary, Alberta
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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

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 1086 - 1096
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Guy, R. K., On a coarseness conjecture of Erdôs, J. Combinatorial Theory 3 (1967), 38–42.Google Scholar
2. Guy, R. K. and Beineke, L. W., The coarseness of the complete graph, Can. J. Math. 20 (1968), 888–894.Google Scholar
3. Kuratowski, K., Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.Google Scholar