The maximum genus of vertex-transitive graphs

Discrete Mathematics, 1991

Skoviera, M., The maximum genus of graphs of diameter two, Discrete Mathematics 87 (1991) 175-180. Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus y,&G) equals [fi(G)/Z], where /3(G) = IF(G)1-IV(G)1 + 1 is the Betti number of G. For graphs with loops we show that [p(G)/21-2s yM(G) c &G)/Z] if G is vertex 2-connected, and compute the exact value of yM(G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [lo] 4-connected graphs are upper embeddable. Theorem 1. Every loopless graph of diameter two is upper embeddable.

1994

Abstract The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

Journal of Graph Theory, 1996

The Petersen graph on 10 vertices is the smallest example of a vertextransitive graph which is not a Cayley graph. In 1983, D. Marušič asked: for what values of n does there exist such a graph on n vertices? We give several new constructions of families of vertex-transitive graphs which are not Cayley graphs and complete the proof that, if n is divisible by p 2 for some prime p, then there is a vertex-transitive graph on n vertices which is not a Cayley graph unless n is p 2 , p 3 , or 12.

Discrete Mathematics, 1976

The vu&age graph construction of Gross is extenSccf to the case where the baw graph is non-orkntably embedded. An easily applied criterion is established for determining the orientability character of the derived embedding. These methods are then applied to derive both orientable ijnd non-orientabte genus embeddings for some families of complete tripartite graphs.

Journal of Combinatorial Theory, Series B, 2006

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 exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.

Discrete Mathematics, 1998

A conjecture of Robertson and Thomas on the orientable genus of graphs with a given nonorientable embedding is disproved.

TEMA (São Carlos), 2016

The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.

Journal of Graph Theory, 2012

In 1983, the second author [D. Marušič, Ars Combinatoria 16B (1983), 297-302] asked for which positive integers n there exists a non-Cayley vertex-transitive graph on n vertices. (The term non-Cayley numbers has later been given to such integers.) Motivated by this problem, Feng [Discrete Math 248 (2002), 265-269] asked to determine the smallest valency ϑ(n) among valencies of non-Cayley vertex-transitive graphs of order n. As cycles are clearly Cayley graphs, ϑ(n) ≥ 3 for any non-Cayley number n. In this paper a goal is set to determine those non-Cayley numbers n for which ϑ(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non-Cayley vertex-transitive Contract grant sponsors: Agencija za raziskovalno dejavnost Republike Slovenije, research program P1-0285 (to K. K., D. M., C. Z.); Agencija za raziskovalno dejavnost Republike Slovenije, proj. mladi raziskovalci (to C. Z.).

2021

A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...