共查询到20条相似文献,搜索用时 15 毫秒
1.
We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive -regular graphs of girth . Counting cycles to obtain necessary arithmetic conditions on the parameters , we extend previous results of Biggs, and prove that, for any given excess and any given degree , the asymptotic density of the set of girths for which there exists a vertex-transitive -cage with excess not exceeding is 0. 相似文献
2.
Jana Tomanová 《Discrete Mathematics》2010,310(1):192-195
We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions. We also prove that the graphs arising this way are hamiltonian provided that their valency is at least six. 相似文献
3.
David Renault 《Discrete Mathematics》2009,309(9):2815-2833
We consider the class of the topologically locally finite (in short TLF) planar vertex-transitive graphs. We characterize these graphs by finite combinatorial objects called labeling schemes. As a result, we are able to enumerate and describe all TLF-planar vertex-transitive graphs of given degree, as well as most of their transitive groups of automorphisms. In addition,we are able to decide whether a given TLF-planar transitive graph is Cayley or not. This class contains all the one-ended planar Cayley graphs and the normal transitive tilings of the plane. 相似文献
4.
A set of vertices in a graph is an independent dominating set of if is an independent set and every vertex not in is adjacent to a vertex in . The independent domination number, , of is the minimum cardinality of an independent dominating set. In this paper, we extend the work of Henning, Löwenstein, and Rautenbach (2014) who proved that if is a bipartite, cubic graph of order and of girth at least , then . We show that the bipartite condition can be relaxed, and prove that if is a cubic graph of order and of girth at least , then . 相似文献
5.
6.
7.
The theory of voltage graphs has become a standard tool in the study of graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting a cyclic group of automorphisms that may not be semiregular. We use this new tool to classify all cubic graphs admitting a cyclic group of automorphisms with at most three vertex-orbits and we characterise vertex-transitivity for each of these classes. In particular, we show that a cubic vertex-transitive graph admitting a cyclic group of automorphisms with at most three orbits on vertices either belongs to one of 5 infinite families or is isomorphic to the well-known Tutte–Coxeter graph. 相似文献
8.
It is shown that every connected vertex-transitive graph of order 6p, where p is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order 6p which is not genuinely imprimitive contains a Hamilton cycle. 相似文献
9.
Ademir Hujdurović 《Journal of Graph Theory》2020,95(4):543-564
A clique (resp, independent set) in a graph is strong if it intersects every maximal independent set (resp, every maximal clique). A graph is clique intersect stable set (CIS) if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique in a vertex-transitive graph is strong if and only if for every maximal independent set of . On the basis of this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transitive graphs of valency at most 4 admitting a strong clique, and give a partial characterization of 5-valent vertex-transitive graphs admitting a strong clique. Our results imply that every vertex-transitive graph of valency at most 5 that admits a strong clique is localizable. We answer an open question by providing an example of a vertex-transitive CIS graph which is not localizable. 相似文献
10.
Let be a planar graph with a list assignment . Suppose a preferred color is given for some of the vertices. We prove that if has girth at least six and all lists have size at least three, then there exists an -coloring respecting at least a constant fraction of the preferences. 相似文献
11.
Eric Mwambene 《Central European Journal of Mathematics》2005,3(2):245-250
Via representation of vertex-transitive graphs on groupoids, we show that left loops with units are factors of groups, i.e.,
left loops are transversals of left cosets on which it is possible to define a binary operation which allows left cancellation. 相似文献
12.
13.
Bahman Khosravi 《Discrete Mathematics》2010,310(4):804-811
In this paper, we first give a characterization of Cayley graphs of rectangular groups. Then, vertex-transitivity of Cayley graphs of rectangular groups is considered. Further, it is shown that Cayley graphs Cay(S,C) which are automorphism-vertex-transitive, are in fact Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox semigroup. Finally, a characterization of vertex-transitive graphs which are Cayley graphs of finite semigroups is concluded. 相似文献
14.
A graph is said to be vertex-transitive non-Cayley if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order 12 p, where p is a prime, is given. As a result, there are 11 sporadic and one infinite family of such graphs, of which the sporadic ones occur when p equals 5, 7 or 17, and the infinite family exists if and only if p ≡ 1(mod 4), and in this family there is a unique graph for a given order. 相似文献
15.
A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549-562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k≥4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k≥4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs. 相似文献
16.
In this article current directions in solving Lovász’s problem about the existence of Hamilton cycles and paths in connected vertex-transitive graphs are given. 相似文献
17.
《Discrete Mathematics》2004,280(1-3):133-148
An infinite family of cubic edge- but not vertex-transitive graphs is constructed. The graphs are obtained as regular
-covers of K3,3 where n=p1e1p2e2pkek where pi are distinct primes congruent to 1 modulo 3, and ei1. Moreover, it is proved that the Gray graph (of order 54) is the smallest cubic edge- but not vertex-transitive graph. 相似文献
18.
19.
《Discrete Mathematics》2022,345(8):112937
We classify trivalent vertex-transitive graphs whose edge sets have a partition into a 2-factor composed of two cycles and a 1-factor that is invariant under the action of the automorphism group. 相似文献
20.
《Discrete Mathematics》2022,345(10):113002
We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974. 相似文献