Additivity of the genus of a graph

Battle, Joseph; Harary, Frank; Kodama, Yuya

doi:10.1090/s0002-9904-1962-10847-7

Cited by 144 publications

(129 citation statements)

References 2 publications

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“…In [5,Corollary 1], it has been proved that the genus of a graph is the sum of the genera of its blocks. Thus, it follows that the following.…”

Section: B∈bmentioning

confidence: 99%

On the Genus of the Commuting Graphs of Finite Non-Abelian Groups

Das¹,

Nongsiang²

2016

International Electronic Journal of Algebra

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Abstract. The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we determine (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of isomorphism classes of finite non-abelian groups whose commuting graphs have the same genus is finite.Mathematics Subject Classification (2010): 20D60, 05C25

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“…In [5,Corollary 1], it has been proved that the genus of a graph is the sum of the genera of its blocks. Thus, it follows that the following.…”

Section: B∈bmentioning

confidence: 99%

On the Genus of the Commuting Graphs of Finite Non-Abelian Groups

Das¹,

Nongsiang²

2016

International Electronic Journal of Algebra

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“…In order to see this we use the result of Battle et al [2] which implies that a graph which contains m disjoint nonplanar subgraphs has genus > m . More precisely, we prove: Lemma 2.4.…”

Section: A ç V(g)ue(g) Then G-a Is the Graph Obtained From G By Delementioning

confidence: 99%

“…By the additivity theorems [2,6] they cannot all be nonplanar if G is large. Thus the omission of the planarity condition would result in only finitely many additional tilings.)…”

Section: A ç V(g)ue(g) Then G-a Is the Graph Obtained From G By Delementioning

confidence: 99%

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Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on a Fixed Surface

Thomassen¹

1991

Transactions of the American Mathematical Society

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Abstract. We describe all regular tilings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each g > 3 , there are only finitely many vertex-transitive graphs of genus g . In fact, they all have order < 101 g . The weaker conjecture for Cayley graphs was made by Gross and Tucker and extends Hurwitz' theorem that, for each g > 2, there are only finitely many groups that act on the surface of genus g . We also derive a nonorientable version of Hurwitz' theorem.

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“…The following upper bound for g(Km V" K ) will be obtained as Proposition 1.15 in §1. E. is probably a poor upper bound, but improving on it seems to be quite difficult.…”

Section: Now Let G H and T Be Graphs Obtained By Amalgamating N Cmentioning

confidence: 99%

The genera of amalgamations of graphs

Alpert¹

1973

Trans. Amer. Math. Soc.

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If p ≤ m p \leq m , n then K m ∨ K p K n {K_m}{ \vee _{{K_p}}}{K_n} is the graph obtained by identify ing a copy of K p {K_p} contained in K m {K_m} with a copy of K p {K_p} contained in K n {K_n} . It is shown that for all integers p ≤ m p \leq m , n the genus g ( K m ∨ K p K n ) g({K_m}{ \vee _{{K_p}}}{K_n}) of K m ∨ K p K n {K_m}{ \vee _{{K_p}}}{K_n} is less than or equal to g ( K m ) + g ( K n ) g({K_m}) + g({K_n}) . Combining this fact with the lower bound obtained from the Euler formula, one sees that for 2 ≤ p ≤ 5 , g ( K m ∨ K p K n ) 2 \leq p \leq 5,g({K_m}{ \vee _{{K_p}}}{K_n}) is either g ( K m ) + g ( K n ) g({K_m}) + g({K_n}) or else g ( K m ) + g ( K n ) − 1 g({K_m}) + g({K_n}) - 1 . Except in a few special cases, it is determined which of these values is actually attained.

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Additivity of the genus of a graph

Cited by 144 publications

References 2 publications

On the Genus of the Commuting Graphs of Finite Non-Abelian Groups

On the Genus of the Commuting Graphs of Finite Non-Abelian Groups

Tilings of the Torus and the Klein Bottle and Vertex-Transitive Graphs on a Fixed Surface

The genera of amalgamations of graphs

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