Counterexamples to the nonorientable genus conjecture for complete tripartite graphs

Authors:
M. N. Ellingham;Chris Stephens;Xiaoya Zha
Affiliations:
Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN;Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nashville, TN;Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN
Venue:
European Journal of Combinatorics - Special issue: Topological graph theory II
Year:
2005

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Abstract

In 1976, Stahl and White conjectured that the minimum nonorientable genus of Kl,m,n (where l ≥ m ≥ n) is ⌈(l-2)(m+n-2)/2⌉. We prove that K4,4,1, K4,4,3, and K3,3,3 are counterexample to this conjecture. We also show that all other complete tripartite graphs Kl,m,n with l ≥ m ≥ n and l ≤ 5 satisfy the conjecture. Moreover, all complete tripartite graphs with l ≤ 5 satisfy the similar conjecture for orientable genus.