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

(1 reference statement)

Supporting

Mentioning

126

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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

show abstract

“…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

View full text Add to dashboard Cite

show abstract

“…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%

See 1 more Smart Citation

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

Thomassen¹

1991

Transactions of the American Mathematical Society

View full text Add to dashboard Cite

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.

show abstract

“…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.

View full text Add to dashboard Cite

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.

show abstract

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

Contact Info

Product

Resources

About