In this paper, we obtain three general reduction formulas to determine the orientable and nonorie... more In this paper, we obtain three general reduction formulas to determine the orientable and nonorientable genera for complete tripartite graphs. As corollaries, we (1) reduce the determination of the orientable (nonorientable, respectively) genera of 75 percent (85 ...
Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are... more Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are equivalent to embedded voltage graphs but are more convenient in certain situations. We describe how transition graphs can be used to construct embeddings and discuss some of their advantages. As applications, we discuss how transition graphs can be used to construct embeddings of complete bipartite
We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are ... more We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, Km¯+Kn=Km+n−Km, and show that for n⩾3 and m⩾n−1 its nonorientable genus
ABSTRACT We prove that if G is a 5-connected graph embedded on a surface Sigma (other than the sp... more ABSTRACT We prove that if G is a 5-connected graph embedded on a surface Sigma (other than the sphere) with face-width at least 5, then G contains a subdivision of K-5. This is a special case of a conjecture of P. Seymour, that every 5-connected nonplanar graph contains a subdivision of K-5. Moreover, we prove that if G is 6-connected and embedded with face-width at least 5, then for every v. V (G), G contains a subdivision of K5 whose branch vertices are v and four neighbors of v. (C) 2012 Wiley Periodicals, Inc.
Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are... more Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are equivalent to embedded voltage graphs but are more convenient in certain situations. We describe how transition graphs can be used to construct ...
In 1976, Stahl and White conjectured that the nonorientable genus of Kl, m, n, where l⩾ m⩾ n, is⌈... more In 1976, Stahl and White conjectured that the nonorientable genus of Kl, m, n, where l⩾ m⩾ n, is⌈(l− 2)(m+ n− 2)/2⌉. The authors recently showed that the graphs K3, 3, 3, K4, 4, 1, and K4, 4, 3 are counterexamples to this conjecture. Here we prove that apart from these three ...
We show that for n= 4 and n⩾ 6, Kn has a nonorientable embedding in which all the facial walks ar... more We show that for n= 4 and n⩾ 6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a ...
In this paper, we obtain three general reduction formulas to determine the orientable and nonorie... more In this paper, we obtain three general reduction formulas to determine the orientable and nonorientable genera for complete tripartite graphs. As corollaries, we (1) reduce the determination of the orientable (nonorientable, respectively) genera of 75 percent (85 ...
Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are... more Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are equivalent to embedded voltage graphs but are more convenient in certain situations. We describe how transition graphs can be used to construct embeddings and discuss some of their advantages. As applications, we discuss how transition graphs can be used to construct embeddings of complete bipartite
We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are ... more We show that for n=4 and n⩾6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a complete graph, Km¯+Kn=Km+n−Km, and show that for n⩾3 and m⩾n−1 its nonorientable genus
ABSTRACT We prove that if G is a 5-connected graph embedded on a surface Sigma (other than the sp... more ABSTRACT We prove that if G is a 5-connected graph embedded on a surface Sigma (other than the sphere) with face-width at least 5, then G contains a subdivision of K-5. This is a special case of a conjecture of P. Seymour, that every 5-connected nonplanar graph contains a subdivision of K-5. Moreover, we prove that if G is 6-connected and embedded with face-width at least 5, then for every v. V (G), G contains a subdivision of K5 whose branch vertices are v and four neighbors of v. (C) 2012 Wiley Periodicals, Inc.
Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are... more Transition graphs are an algebraic way of constructing embeddings of graphs in surfaces. They are equivalent to embedded voltage graphs but are more convenient in certain situations. We describe how transition graphs can be used to construct ...
In 1976, Stahl and White conjectured that the nonorientable genus of Kl, m, n, where l⩾ m⩾ n, is⌈... more In 1976, Stahl and White conjectured that the nonorientable genus of Kl, m, n, where l⩾ m⩾ n, is⌈(l− 2)(m+ n− 2)/2⌉. The authors recently showed that the graphs K3, 3, 3, K4, 4, 1, and K4, 4, 3 are counterexamples to this conjecture. Here we prove that apart from these three ...
We show that for n= 4 and n⩾ 6, Kn has a nonorientable embedding in which all the facial walks ar... more We show that for n= 4 and n⩾ 6, Kn has a nonorientable embedding in which all the facial walks are hamilton cycles. Moreover, when n is odd there is such an embedding that is 2-face-colorable. Using these results we consider the join of an edgeless graph with a ...
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