共查询到20条相似文献，搜索用时 15 毫秒
1.
Dragan Maru&#x;i
《Journal of Graph Theory》2000,35(2):152-160
An infinite family of cubic edge‐transitive but not vertex‐transitive graphs with edge stabilizer isomorphic to ℤ2 is constructed. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 152–160, 2000 相似文献
2.
A graph is vertex?transitive or symmetric if its automorphism group acts transitively on vertices or ordered adjacent pairs of vertices of the graph, respectively. Let G be a finite group and S a subset of G such that 1?S and S={s?1 | s∈S}. The Cayleygraph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, sg} | g∈G, s∈S}. Feng and Kwak [J Combin Theory B 97 (2007), 627–646; J Austral Math Soc 81 (2006), 153–164] classified all cubic symmetric graphs of order 4p or 2p2 and in this article we classify all cubic symmetric graphs of order 2pq, where p and q are distinct odd primes. Furthermore, a classification of all cubic vertex‐transitive non‐Cayley graphs of order 2pq, which were investigated extensively in the literature, is given. As a result, among others, a classification of cubic vertex‐transitive graphs of order 2pq can be deduced. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 285–302, 2010 相似文献
3.
Jin‐Xin Zhou 《Journal of Graph Theory》2012,71(4):402-415
A graph is vertex‐transitive if its automorphism group acts transitively on vertices of the graph. A vertex‐transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this article, the tetravalent vertex‐transitive non‐Cayley graphs of order 4p are classified for each prime p. As a result, there are one sporadic and five infinite families of such graphs, of which the sporadic one has order 20, and one infinite family exists for every prime p>3, two families exist if and only if p≡1 (mod 8) and the other two families exist if and only if p≡1 (mod 4). For each family there is a unique graph for a given order. © 2011 Wiley Periodicals, Inc. 相似文献
4.
In 1983, the second author [D. Maru?i?, 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 ?(n) among valencies of non‐Cayley vertex‐transitive graphs of order n. As cycles are clearly Cayley graphs, ?(n)?3 for any non‐Cayley number n. In this paper a goal is set to determine those non‐Cayley numbers n for which ?(n) = 3, and among the latter to determine those for which the generalized Petersen graphs are the only non‐Cayley vertex‐transitive graphs of order n. It is known that for a prime p every vertex‐transitive graph of order p, p2 or p3 is a Cayley graph, and that, with the exception of the Coxeter graph, every cubic non‐Cayley vertex‐transitive graph of order 2p, 4p or 2p2 is a generalized Petersen graph. In this paper the next natural step is taken by proving that every cubic non‐Cayley vertex‐transitive graph of order 4p2, p>7 a prime, is a generalized Petersen graph. In addition, cubic non‐Cayley vertex‐transitive graphs of order 2pk, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012 相似文献
5.
Given natural numbers n?3 and 1?a, r?n?1, the rose window graph Rn(a, r) is a quartic graph with vertex set ${{{x}}_{{i}}|{{i}}in {mathbb{Z}}_{{n}}} cup {{{y}}_{{i}}|{{i}}in{mathbb{Z}}_{{n}}}Given natural numbers n?3 and 1?a, r?n?1, the rose window graph Rn(a, r) is a quartic graph with vertex set ${{{x}}_{{i}}|{{i}}in {mathbb{Z}}_{{n}}} cup {{{y}}_{{i}}|{{i}}in{mathbb{Z}}_{{n}}}$ and edge set ${{{{x}}_{{i}},{{x}}_{{{i+1}}}} mid {{i}}in {mathbb{Z}}_n } cup {{{{y}}_{{{i}}},{{y}}_{{{i+r}}}}mid {{i}} in{mathbb{Z}}_{{n}}}cup {{{{x}}_{{{i}}},{{y}}_{{{i}}}} mid {{i}}in {mathbb{Z}}_{{{n}}}}cup {{{{x}}_{{{i+a}}},{{y}}_{{{i}}}} mid{{i}} in {mathbb{Z}}_{{{n}}}}$. In this article a complete classification of edge‐transitive rose window graphs is given, thus solving one of the three open problems about these graphs posed by Steve Wilson in 2001. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 216–231, 2010 相似文献
6.
设图G是一个K-正则连通点可迁图.如果G不是极大限制性边连通的,那么G含有一个(k-1)-因子,它的所有分支都同构于同一个阶价于k和2k-3之间的点可迁图.此结果在某种程度上加强了Watkins的相应命题:如果k正则点可迁图G不是k连通的,那么G有一个因子,它的每一个分支都同构于同一个点可迁图. 相似文献
7.
For any d?5 and k?3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d?3)/3)k. By comparison with other available results in this area we show that our family gives the largest currently known Cayley graphs for a wide range of sufficiently large degrees and diameters. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 87–98, 2010 相似文献
8.
A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p). 相似文献
9.
We classify noncomplete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of 2‐geodesics. We prove that either Γ is 2‐arc transitive or the valency p satisfies , and for each such prime there is a unique graph with this property: it is a nonbipartite antipodal double cover of the complete graph with automorphism group and diameter 3. 相似文献
10.
The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gn)/|V(Gn)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory 相似文献
11.
Let G be a connected, nonbipartite vertex‐transitive graph. We prove that if the only independent sets of maximal cardinality in the tensor product G × G are the preimages of the independent sets of maximal cardinality in G under projections, then the same holds for all finite tensor powers of G, thus providing an affirmative answer to a question raised by Larose and Tardif (J Graph Theory 40(3) (2002), 162–171). © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 295‐301, 2009 相似文献
12.
A d‐regular graph is said to be superconnected if any disconnecting subset with cardinality at most d is formed by the neighbors of some vertex. A superconnected graph that remains connected after the failure of a vertex and its neighbors will be called vosperian. Let Γ be a vertex‐transitive graph of degree d with order at least d+4. We give necessary and sufficient conditions for the vosperianity of Γ. Moreover, assuming that distinct vertices have distinct neighbors, we show that Γ is vosperian if and only if it is superconnected. Let G be a group and let S?G{1} with S=S?1. We show that the Cayley graph, Cay(G, S), defined on G by S is vosperian if and only if G(S∪{1}) is not a progression and for every non‐trivial subgroup H and every a∈G, If moreover S is aperiodic, then Cay(G, S) is vosperian if and only if it is superconnected. © 2011 Wiley Periodicals, Inc. J Graph Theory 67:124‐138, 2011 相似文献
13.
For every d and k, we determine the smallest order of a vertex‐transitive graph of degree d and diameter k, and in each such case we show that this order is achieved by a Cayley graph. 相似文献
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A graph of order n is p ‐factor‐critical, where p is an integer of the same parity as n, if the removal of any set of p vertices results in a graph with a perfect matching. 1‐factor‐critical graphs and 2‐factor‐critical graphs are factor‐critical graphs and bicritical graphs, respectively. It is well known that every connected vertex‐transitive graph of odd order is factor‐critical and every connected nonbipartite vertex‐transitive graph of even order is bicritical. In this article, we show that a simple connected vertex‐transitive graph of odd order at least five is 3‐factor‐critical if and only if it is not a cycle. 相似文献
16.
Mader and Jackson independently proved that every 2‐connected simple graph G with minimum degree at least four has a removable cycle, that is, a cycle C such that G/E(C) is 2‐connected. This paper considers the problem of determining when every edge of a 2‐connected graph G, simple or not, can be guaranteed to lie in some removable cycle. The main result establishes that if every deletion of two edges from G remains 2‐connected, then, not only is every edge in a removable cycle but, for every two edges, there are edge‐disjoint removable cycles such that each contains one of the distinguished edges. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 155–164, 2003 相似文献
17.
Mader proved that every 2‐connected simple graph G with minimum degree d exceeding three has a cycle C, the deletion of whose edges leaves a 2‐connected graph. Jackson extended this by showing that C may be chosen to avoid any nominated edge of G and to have length at least d − 1. This article proves an extension of Jackson's theorem. In addition, a conjecture of Goddyn, van den Heuvel, and McGuinness is disproved when it is shown that a natural matroid dual of Mader's theorem fails. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 51–66, 1999 相似文献
18.
图X称为弱点传递图如果X的自同态幺半群EndX在顶点集V(X)上的作用是传递的 .本文给出了广义Petersen图是二分图的充要条件 ,刻划了奇围长小于 9的广义Petersen图的弱点传递性 ,作为推论给出了所有h ≤ 1 5的弱点传递的广义Pe tersen图P(h ,t) . 相似文献
19.
J. Díaz A. C. Kaporis G. D. Kemkes L. M. Kirousis X. Pérez N. Wormald 《Journal of Graph Theory》2009,61(3):157-191
It was only recently shown by Shi and Wormald, using the differential equation method to analyze an appropriate algorithm, that a random 5‐regular graph asymptotically almost surely has chromatic number at most 4. Here, we show that the chromatic number of a random 5‐regular graph is asymptotically almost surely equal to 3, provided a certain four‐variable function has a unique maximum at a given point in a bounded domain. We also describe extensive numerical evidence that strongly suggests that the latter condition holds. The proof applies the small subgraph conditioning method to the number of locally rainbow balanced 3‐colorings, where a coloring is balanced if the number of vertices of each color is equal, and locally rainbow if every vertex is adjacent to at least one vertex of each of the other colors. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 157–191, 2009 相似文献
20.
We study a family of digraphs (directed graphs) that generalises the class of Cayley digraphs. For nonempty subsets of a group G, we define the two‐sided group digraph to have vertex set G, and an arc from x to y if and only if for some and . In common with Cayley graphs and digraphs, two‐sided group digraphs may be useful to model networks as the same routing and communication scheme can be implemented at each vertex. We determine necessary and sufficient conditions on L and R under which may be viewed as a simple graph of valency , and we call such graphs two‐sided group graphs. We also give sufficient conditions for two‐sided group digraphs to be connected, vertex‐transitive, or Cayley graphs. Several open problems are posed. Many examples are given, including one on 12 vertices with connected components of sizes 4 and 8. 相似文献