On cubic non‐Cayley vertex‐transitive graphs

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 2p^k, where p>7 is a prime and k?p, are characterized. © 2011 Wiley Periodicals, Inc. J Graph Theory 69: 77–95, 2012     

Tetravalent vertex‐transitive graphs of order 4p

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.     

Vertex‐Transitive Graphs of Prime‐Squared Order Are Hamilton‐Decomposable

We prove that all connected vertex‐transitive graphs of order p², p a prime, can be decomposed into Hamilton cycles.     

Circulant Double Coverings of a Circulant Graph of Valency Four

Several isomorphism classes of graph coverings of a graph G have been enumerated by many authors (see [3], [8]–[15]). A covering of G is called circulant if its covering graph is circulant. Recently, the authors [4] enumerated the isomorphism classes of circulant double coverings of a certain kind, called typical, and showed that no double covering of a circulant graph of valency 3 is circulant. In this paper, the isomorphism classes of connected circulant double coverings of a circulant graph of valency 4 are enumerated. As a consequence, it is shown that no double covering of a non-circulant graph G of valency 4 can be circulant if G is vertex-transitive or G has a prime power of vertices. The first author is supported by NSF of China (No. 60473019) and by NKBRPC (2004CB318000), and the second author is supported by Com²MaC-KOSEF (R11-1999-054) in Korea.     

3‐Factor‐Criticality of Vertex‐Transitive Graphs

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.     

On the Classification of Arc-transitive Circulant Digraphs of Order Odd-Prime-Squared

A Cayley graph F = Cay（G, S） of a group G with respect to S is called a circulant digraph of order pk if G is a cyclic group of the same order. Investigated in this paper are the normality conditions for arc-transitive circulant （di）graphs of order p^2 and the classification of all such graphs. It is proved that any connected arc-transitive circulant digraph of order p^2 is, up to a graph isomorphism, either Kp2, G（p^2,r）, or G（p,r）[pK1], where r｜p- 1.     

Cubic vertex‐transitive graphs of order 2pq

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     

On semisymmetric cubic graphs of order 6p~2

A graph r is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut(Г) acting transitively on its edge set but not on its vertex set. In the case of G. = A, we call r a semisymmetric graph. The aim of this paper is to investigate (G-)semisymmetric graphs of prime degree. We give a group-theoretical construction of such graphs, and give a classification of semisymmetric cubic graphs of order 6p2 for an odd prime p.     

Circulant Double Coverings of a Circulant Graph of Valency Five

Enumerating the isomorphism classes of several types of graph covering projections is one of the central research topics in enumerative topological graph theory. A covering of G is called circulant if its covering graph is circulant. Recently, the authors [Discrete Math., 277, 73-85 （2004）1 enumerated the isomorphism classes of circulant double coverings of a certain type, called a typical covering, and showed that no double covering of a circulant graph of valency three is circulant. Also, in [Graphs and Combinatorics, 21,386 400 （2005）], the isomorphism classes of circulant double coverings of a circulant graph of valency four are enumerated. In this paper, the isomorphism classes of circulant double coverings of a circulant graph of valency five are enumerated.     

Some Identities for Enumerators of Circulant Graphs

We establish analytically several new identities connecting enumerators of different types of circulant graphs mainly of prime, twice prime and prime-squared orders. In particular, it is shown that the half-sum of the number of undirected circulants and the number of undirected self-complementary circulants of prime order is equal to the number of directed self-complementary circulants of the same order. Several identities hold only for prime orders p such that (p + 1)/2 is also prime. Some conjectured generalizations and interpretations are discussed.     

Pseudo‐random properties of self‐complementary symmetric graphs

There are some results in the literature showing that Paley graphs behave in many ways like random graphs G(n, 1/2). In this paper, we extend these results to the other family of self‐complementary symmetric graphs. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 310–316, 2004     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

Perfect 1‐Factorizations of a Family of Cayley Graphs

A 1‐factorization of a graph G is a decomposition of G into edge‐disjoint 1‐factors (perfect matchings), and a perfect 1‐factorization is a 1‐factorization in which the union of any two of the 1‐factors is a Hamilton cycle. We consider the problem of the existence of perfect 1‐factorizations of even order 4‐regular Cayley graphs, with a particular interest in circulant graphs. In this paper, we study a new family of graphs, denoted , which are Cayley graphs if and only if k is even or . By solving the perfect 1‐factorization problem for a large class of graphs, we obtain a new class of 4‐regular bipartite circulant graphs that do not have a perfect 1‐factorization, answering a problem posed in 7 . With further study of graphs, we prove the nonexistence of P1Fs in a class of 4‐regular non‐bipartite circulant graphs, which is further support for a conjecture made in 7 .     

Vertex‐Transitive Cubic Graphs of Square‐Free Order

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).     

Decomposing complete equipartite graphs into odd square‐length cycles: number of parts odd

In this article, we introduce a new technique for obtaining cycle decompositions of complete equipartite graphs from cycle decompositions of related multigraphs. We use this technique to prove that if n, m and λ are positive integers with n ≥ 3, λ≥ 3 and n and λ both odd, then the complete equipartite graph having n parts of size m admits a decomposition into cycles of length λ² whenever nm ≥ λ² and λ divides m. As a corollary, we obtain necessary and sufficient conditions for the decomposition of any complete equipartite graph into cycles of length p², where p is prime. © 2010 Wiley Periodicals, Inc. J Combin Designs 18:401‐414, 2010     

Tetravalent half‐edge‐transitive graphs and non‐normal Cayley graphs

Let X be a vertex‐transitive graph, that is, the automorphism group Aut(X) of X is transitive on the vertex set of X. The graph X is said to be symmetric if Aut(X) is transitive on the arc set of X. suppose that Aut(X) has two orbits of the same length on the arc set of X. Then X is said to be half‐arc‐transitive or half‐edge‐transitive if Aut(X) has one or two orbits on the edge set of X, respectively. Stabilizers of symmetric and half‐arc‐transitive graphs have been investigated by many authors. For example, see Tutte [Canad J Math 11 (1959), 621–624] and Conder and Maru?i? [J Combin Theory Ser B 88 (2003), 67–76]. It is trivial to construct connected tetravalent symmetric graphs with arbitrarily large stabilizers, and by Maru?i? [Discrete Math 299 (2005), 180–193], connected tetravalent half‐arc‐transitive graphs can have arbitrarily large stabilizers. In this article, we show that connected tetravalent half‐edge‐transitive graphs can also have arbitrarily large stabilizers. A Cayley graph Cay(G, S) on a group G is said to be normal if the right regular representation R(G) of G is normal in Aut(Cay(G, S)). There are only a few known examples of connected tetravalent non‐normal Cayley graphs on non‐abelian simple groups. In this article, we give a sufficient condition for non‐normal Cayley graphs and by using the condition, infinitely many connected tetravalent non‐normal Cayley graphs are constructed. As an application, all connected tetravalent non‐normal Cayley graphs on the alternating group A₆ are determined. © 2011 Wiley Periodicals, Inc. J Graph Theory     

On almost self-complementary graphs

A graph is called almost self-complementary if it is isomorphic to one of its almost complements X^c-I, where X^c denotes the complement of X and I a perfect matching (1-factor) in X^c. Almost self-complementary circulant graphs were first studied by Dobson and Šajna [Almost self-complementary circulant graphs, Discrete Math. 278 (2004) 23-44]. In this paper we investigate some of the properties and constructions of general almost self-complementary graphs. In particular, we give necessary and sufficient conditions on the order of an almost self-complementary regular graph, and construct infinite families of almost self-complementary regular graphs, almost self-complementary vertex-transitive graphs, and non-cyclically almost self-complementary circulant graphs.     

Inequalities for determinants and permanents

Let p be a prime > 3. It is shown that no integral circulant of order p^k exists with determinant p^k+1 . It is also shown that m is the determinant of an integral 9×9 circulant if and only if (m, 3)=l, or m = 0 mod 27. The proof makes use of a criterion which must be satisfied by the difference of two units in the cyclotomic field of level p^k .     

A classification of cubic symmetric graphs of order 16<Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

A graph is called symmetric if its automorphism group acts transitively on its arc set. In this paper, we classify all connected cubic symmetric graphs of order 16p ² for each prime p.     

(2 + ϵ)‐Coloring of planar graphs with large odd‐girth

The odd‐girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function ƒ(ϵ) for each ϵ : 0 < ϵ < 1 such that, if the odd‐girth of a planar graph G is at least ƒ(ϵ), then G is (2 + ϵ)‐colorable. Note that the function ƒ(ϵ) is independent of the graph G and ϵ → 0 if and only if ƒ(ϵ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd‐girth of planar graphs. This lemma is expected to have applications in related problems. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 109–119, 2000