Pentavalent One-regular Graphs of Square-free Order

2010 ◽  
Vol 17 (03) ◽  
pp. 515-524 ◽  
Author(s):  
Yantao Li ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Regular Graph ◽  
Cayley Graphs ◽  
Regular Graphs ◽  
Dihedral Groups ◽  
Free Order

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. Let n be a square-free integer. It is shown in this paper that a pentavalent one-regular graph of order n exists if and only if n = 2 · 5tp1p2 … ps ≥ 62, where t ≤ 1, s ≥ 1, and pi's are distinct primes such that 5|(pi-1). For such an integer n, there are exactly 4s-1 non-isomorphic pentavalent one-regular graphs of order n, which are Cayley graphs on dihedral groups constructed by Kwak et al. This work is a continuation of the classification of cubic one-regular graphs of order twice a square-free integer given by Zhou and Feng.

Arc-Transitive Graphs of Square-free Order with Valency 11

2021 ◽  
Vol 28 (04) ◽  
pp. 645-654
Author(s):  
Guang Li ◽  
Bo Ling ◽  
Zaiping Lu
Keyword(s):  
Simple Group ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Cayley Graphs ◽  
Complete List ◽  
Dihedral Groups ◽  
Transitive Graphs ◽  
Free Order ◽  
Normal Cayley Graphs

In this paper, we present a complete list of connected arc-transitive graphs of square-free order with valency 11. The list includes the complete bipartite graph [Formula: see text], the normal Cayley graphs of dihedral groups and the graphs associated with the simple group [Formula: see text] and [Formula: see text], where [Formula: see text] is a prime.

s-Arc-regular prime-valent Cayley graphs of square-free order

2019 ◽  
Vol 18 (05) ◽  
pp. 1950092
Author(s):  
Jiangmin Pan ◽  
Menglin Ding ◽  
Bo Ling
Keyword(s):  
Cayley Graphs ◽  
Complete List ◽  
Regular Graphs ◽  
Free Order

A high cited paper [Feng and Li, One-regular graphs of square-free order of prime valency, Europ. J. Combin. 32 (2011) 261–275] classified 1-arc-regular prime-valent graphs (necessarily Cayley graphs) of square-free order. It is thus interesting to ask what are the [Formula: see text]-arc-regular prime-valent Cayley graphs of square-free order for each [Formula: see text]. In this paper, a complete list of all such graphs is given. The proving process also involves a complete list of the vertex stabilizers of [Formula: see text]-arc-regular prime-valent graphs of any order.

scholarly journals One-regular cubic graphs of order a small number times a prime or a prime square

2004 ◽  
Vol 76 (3) ◽  
pp. 345-356 ◽  
Author(s):  
Yan-Quan Feng ◽  
Jin Ho Kwak
Keyword(s):  
Automorphism Group ◽  
Cayley Graphs ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Dihedral Groups ◽  
Fixed Order

AbstractA graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p2 where p is a prime if and only if 3 is a divisor of p – 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p2.

scholarly journals CLASSIFICATION OF TETRAVALENT -TRANSITIVE NONNORMAL CAYLEY GRAPHS OF FINITE SIMPLE GROUPS

2021 ◽  
pp. 1-9
Author(s):  
XIN GUI FANG ◽  
JIE WANG ◽  
SANMING ZHOU
Keyword(s):  
Automorphism Group ◽  
Simple Group ◽  
Cayley Graph ◽  
Finite Simple Group ◽  
Cayley Graphs ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Edge Transitive

Abstract A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , $\text{M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

scholarly journals Decomposition of (2k+1)-regular graphs containing special spanning 2k-regular Cayley graphs into paths of length 2k+1

2020 ◽  
Author(s):  
Fábio Botler ◽  
Luiz Hoffmann
Keyword(s):  
Regular Graph ◽  
Perfect Matching ◽  
Cayley Graphs ◽  
Regular Graphs

A Pl-decomposition of a graph G is a set of paths with l edges in G that cover the edge set of G. Favaron, Genest, and Kouider (2010) conjectured that every (2k+1)-regular graph that contains a perfect matching admits a P2k+1-decomposition. They also verified this conjecture for 5-regular graphs without cycles of length 4. In 2015, Botler, Mota, and Wakabayashi extended this result to 5-regular graphs without triangles. In this paper, we verify this conjecture for (2k+1)-regular graphs that contain the k-th power of a spanning cycle; and for 5-regular graphs that contain certain spanning 4-regular Cayley graphs.

scholarly journals Which Haar graphs are Cayley graphs?

10.37236/5240 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
István Estélyi ◽  
Tomaž Pisanski
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Cayley Graph ◽  
Regular Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Equivalent Condition ◽  
Dihedral Groups ◽  
A Finite Group

For a finite group $G$ and subset $S$ of $G,$ the Haar graph $H(G,S)$ is a bipartite regular graph, defined as a regular $G$-cover of a dipole with $|S|$ parallel arcs labelled by elements of $S$. If $G$ is an abelian group, then $H(G,S)$ is well-known to be a Cayley graph; however, there are examples of non-abelian groups $G$ and subsets $S$ when this is not the case. In this paper we address the problem of classifying finite non-abelian groups $G$ with the property that every Haar graph $H(G,S)$ is a Cayley graph. An equivalent condition for $H(G,S)$ to be a Cayley graph of a group containing $G$ is derived in terms of $G, S$ and $\mathrm{Aut } G$. It is also shown that the dihedral groups, which are solutions to the above problem, are $\mathbb{Z}_2^2,D_3,D_4$ and $D_{5}$. 

scholarly journals The Classification of $2$-Extendable Edge-Regular Graphs with Diameter $2$

10.37236/8073 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Klavdija Kutnar ◽  
Dragan Marušič ◽  
Štefko Miklavič ◽  
Primož Šparl
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Perfect Matching ◽  
Negative Integer ◽  
Regular Graphs ◽  
Even Order

Let $\ell$ denote a non-negative integer. A connected graph $\Gamma$ of even order at least $2\ell+2$ is $\ell$-extendable if it contains a matching of size $\ell$ and if every such matching is contained in a perfect matching of $\Gamma$. A connected regular graph $\Gamma$ is edge-regular, if there exists an integer $\lambda$ such that every pair of adjacent vertices of $\Gamma$ have exactly $\lambda$  common neighbours. In this paper we classify $2$-extendable edge-regular graphs of even order and diameter $2$.

scholarly journals Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Mehdi Alaeiyan ◽  
Siamak Firouzian ◽  
Mohsen Ghasemi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Partial Answer ◽  
Dihedral Groups ◽  
Undirected Edge ◽  
A Finite Group ◽  
Edge Transitive

A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).

scholarly journals On Minimal Distance-Regular Cayley Graphs of Generalized Dihedral Groups

10.37236/9755 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Štefko Miklavič ◽  
Primož Šparl
Keyword(s):  
Closed Subset ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Complete Classification ◽  
Minimal Distance ◽  
Dihedral Groups ◽  
Generalized Dihedral Group

Let $G$ denote a finite generalized dihedral group with identity $1$ and let $S$ denote an inverse-closed subset of $G \setminus \{1\}$, which generates $G$ and for which there exists $s \in S$, such that $\langle S \setminus \{s,s^{-1}\} \rangle \ne G$. In this paper we obtain the complete classification of distance-regular Cayley graphs $\mathrm{Cay}(G;S)$ for such pairs of $G$ and $S$.

scholarly journals Cycle partitions of regular graphs

2020 ◽  
pp. 1-24
Author(s):  
Vytautas Gruslys ◽  
Shoham Letzter
Keyword(s):  
Regular Graph ◽  
Bipartite Graphs ◽  
Disjoint Paths ◽  
Regular Graphs ◽  
Simple Construction ◽  
Vertex Set ◽  
Almost All ◽  
Vertex Disjoint

Abstract Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into $n / (d+1)$ paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when $d = \Omega(n)$ , improving a result of Han, who showed that in this range almost all vertices of G can be covered by $n / (d+1) + 1$ vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if $d = \Omega(n)$ and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).

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