The Thickness of Some Complete Bipartite and Tripartite Graphs

In this paper, we obtain the thickness for some complete k–partite graphs for k = 2, 3. We first compute the thickness of K_n,n+8 by giving a planar decomposition of K_4k−1,4k+7 for k ≥ 3. Then, two planar decompositions for K_1,g,g(g−1) when g is even and for \(K_{1,g,{1\over{2}}(g-1)^{2}}\) $$K_{1,g,\frac{1}{2}(g-1{)}^2}$$ when g is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of K_m,n is \(\lceil {mn\over{2(m+n-2)}}\rceil\) $$\lceil \frac{mn}{2(m+n-2)}\rceil$$ for any positive integers m, n.