Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

Cube factorizations of complete graphs

A cube factorization of the complete graph on n vertices, K_n, is a 3‐factorization of K_n in which the components of each factor are cubes. We show that there exists a cube factorization of K_n if and only if n ≡ 16 (mod 24), thus providing a new family of uniform 3‐factorizations as well as a partial solution to an open problem posed by Kotzig in 1979. © 2004 Wiley Periodicals, Inc.     

Applications of balance equations to vertex switching reconstruction

A graph is called s-vertex switching reconstructible (s-VSR) if it is uniquely defined, up to isomorphism, by the multiset of unlabeled graphs obtained by switching of all its s-vertex subsets. We show that a graph with n vertices is n/2-VSR if n = 0(mod 4), (n ? 2)/2-VSR and n/2-VSR if n = 2(mod 4), (n ? 1)/2-VSR if n = 1 (mod 2). For hypothetical nonreconstructible graphs, we give bounds on the number of edges (for any s) and on the maximum and minimum degree (for s = 2). We also show that for n > 9 the degree sequence is 2-VSR.     

Q5‐factorization of λKn

Q_k is the simple graph whose vertices are the k‐tuples with entries in {0, 1} and edges are the pairs of k‐tuples that differ in exactly one position. In this paper, we proved that there exists a Q₅‐factorization of λK_n if and only if (a) n ≡ 0(mod 32) if λ ≡ 0(mod 5) and (b) n ≡ 96(mod 160) if λ ? 0(mod 5).     

Chromaticity of two-trees

The graphs called 2-trees are defined by recursion. The smallest 2-tree is the complete graph on 2 vertices. A 2-tree on n + 1 vertices (where n ≥ 2) is obtained by adding a new vertex adjacent to each of 2 arbitrarily selected adjacent vertices in a 2-tree on n vertices. A graph G is a 2-tree on n(≥2) vertices if and only if its chromatic polynomial is equal to γ(γ - 1)(γ - 2)^n—2.     

Non-Hamiltonian non-Grinbergian graphs

Settling a question of Tutte and a similar question of Grünbaum and Zaks, we present a 3-valent 3-connected planar graph that has only pentagons and octagons, has 92 (200, respectively) vertices and its longest circuit (path, respectively) contains at most 90 (198, respectively) vertices; moreover, it is shown that the family of all 3-valent 3-connected planar graphs, having n-gons only if n≡ 2 (mod3) (or n≡ 1 (mod3)), has a shortness exponent which is less than one.     

Complete subgraphs with large degree sums

Let t(n, k) denote the Turán number—the maximum number of edges in a graph on n vertices that does not contain a complete graph K_k+1. It is shown that if G is a graph on n vertices with n ≥ k²(k – 1)/4 and m < t(n, k) edges, then G contains a complete subgraph K_k such that the sum of the degrees of the vertices is at least 2km/n. This result is sharp in an asymptotic sense in that the sum of the degrees of the vertices of K_k is not in general larger, and if the number of edges in G is at most t(n, k) – ? (for an appropriate ?), then the conclusion is not in general true. © 1992 John Wiley & Sons, Inc.     

On the Structure of Graphs with Low Obstacle Number

The obstacle number of a graph G is the smallest number of polygonal obstacles in the plane with the property that the vertices of G can be represented by distinct points such that two of them see each other if and only if the corresponding vertices are joined by an edge. We list three small graphs that require more than one obstacle. Using extremal graph theoretic tools developed by Pr?mel, Steger, Bollobás, Thomason, and others, we deduce that for any fixed integer h, the total number of graphs on n vertices with obstacle number at most h is at most 2^o(n2){2^{o(n^2)}}. This implies that there are bipartite graphs with arbitrarily large obstacle number, which answers a question of Alpert et al. (Discret Comput Geom doi:, 2009).     

Ramsey‐type results for Gallai colorings

A Gallai‐coloring of a complete graph is an edge coloring such that no triangle is colored with three distinct colors. Gallai‐colorings occur in various contexts such as the theory of partially ordered sets (in Gallai's original paper) or information theory. Gallai‐colorings extend 2‐colorings of the edges of complete graphs. They actually turn out to be close to 2‐colorings—without being trivial extensions. Here, we give a method to extend some results on 2‐colorings to Gallai‐colorings, among them known and new, easy and difficult results. The method works for Gallai‐extendible families that include, for example, double stars and graphs of diameter at most d for 2?d, or complete bipartite graphs. It follows that every Gallai‐colored K_n contains a monochromatic double star with at least 3n+ 1/4 vertices, a monochromatic complete bipartite graph on at least n/2 vertices, monochromatic subgraphs of diameter two with at least 3n/4 vertices, etc. The generalizations are not automatic though, for instance, a Gallai‐colored complete graph does not necessarily contain a monochromatic star on n/2 vertices. It turns out that the extension is possible for graph classes closed under a simple operation called equalization. We also investigate Ramsey numbers of graphs in Gallai‐colorings with a given number of colors. For any graph H let RG(r, H) be the minimum m such that in every Gallai‐coloring of K_m with r colors, there is a monochromatic copy of H. We show that for fixed H, RG (r, H) is exponential in r if H is not bipartite; linear in r if H is bipartite but not a star; constant (does not depend on r) if H is a star (and we determine its value). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 233–243, 2010     

Large induced forests in sparse graphs

For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree Δ. Our results include: (a) if Δ ≤ 3, and G ≠ K₄, then a(G) ≥ n ? e/4 ? 1/4 and this is sharp for all permissible e ≡ 3 (mod 4); and (b) if Δ ≥ 3, then a(G) ≥ α(G) + (n ? α(G))/(Δ ? 1)². Several problems remain open. © 2001 John Wiley & Sons, Inc. J Graph Theory 38: 113–123, 2001     

On the Maximum Number of Cliques in a Graph

A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for the following graph classes: (1) graphs with n vertices and m edges; (2) graphs with n vertices, m edges, and maximum degree Δ; (3) d-degenerate graphs with n vertices and m edges; (4) planar graphs with n vertices and m edges; and (5) graphs with n vertices and no K₅-minor or no K_3,3-minor. For example, the maximum number of cliques in a planar graph with n vertices is 8(n − 2). Research supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen. Cat 2001SGR00224.     

Induced Decompositions of Graphs

We consider those graphs G that admit decompositions into copies of a fixed graph F, each copy being an induced subgraph of G. We are interested in finding the extremal graphs with this property, that is, those graphs G on n vertices with the maximum possible number of edges. We discuss the cases where F is a complete equipartite graph, a cycle, a star, or a graph on at most four vertices.     

Unavoidable Configurations in Complete Topological Graphs

A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least clog^1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a non-crossing subgraph isomorphic to any fixed tree with at most clog^1/6 n vertices.     

Cubic vertex-transitive non-Cayley graphs of order 12<Emphasis Type="Italic">p</Emphasis>

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 12p, 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.     

Automorphisms of coverings of strongly regular graphs with parameters (81,20,1,6)

The possible orders and subgraphs of fixed points of the automorphisms of distance-regular graphs of diameter 4 which are r-coverings of a strongly regular graph with parameters (81, 20, 1, 6) for r ∈ {2, 3, 6} are found. It is proved that, if the automorphism group of a covering of the above type acts transitively on the set of vertices of the graph, then r = 3.     

Automorphisms of random graphs with specified vertices

Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1. These conditions involve the probability that such a graph has a given subgraph. One implication is that the probability that a random unlabelledk-regular simple graph onn vertices has only the trivial group of automorphisms is asymptotic to 1 asn → ∞ with 3≦k=O(n ^1/2−c). In combination with previously known results, this produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymptotic results on the probable connectivity and girth of such graphs. Corresponding results for graphs with more arbitrary degree sequences are obtained. The main results apply equally well to graphs in which multiple edges and loops are permitted, and also to bicoloured graphs. Research of the second author supported by U. S. National Science Foundation Grant MCS-8101555, and by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowships Scheme. Current address: Mathematics Department, University of Auckland, Auckland, New Zealand.     

Triangles in self-complementary graphs

It is shown that the number of triangles in a self-complementary graph with N vertices is at least $\text{N(N ? 2)(N ? 4)}\text{48}$ if N ≡ 0 (mod 4) and at least $\text{N(N ? 1)(N ? 5)}\text{48}$ if N ≡ 1 (mod 4), and that this minimum number can be achieved.     

On the Ramsey number of trees versus graphs with large clique number

Chvátal established that r(T_m, K_n) = (m – 1)(n – 1) + 1, where T_m is an arbitrary tree of order m and K_n is the complete graph of order n. This result was extended by Chartrand, Gould, and Polimeni who showed K_n could be replaced by a graph with clique number n and order n + 1 provided n ≧ 3 and m ≧ 3. We further extend these results to show that K_n can be replaced by any graph on n + 2 vertices with clique number n, provided n ≧ 5 and m ≧ 4. We then show that further extensions, in particular to graphs on n + 3 vertices with clique number n are impossible. We also investigate the Ramsey number of trees versus complete graphs minus sets of independent edges. We show that r(T_m, K_n –tK₂) = (m – 1)(n – t – 1) + 1 for m ≧ 3, n ≧ 6, where T_m is any tree of order m except the star, and for each t, O ≦ t ≦ [(n – 2)/2].     

Positive integers expressible as a sum of three squares in essentially only one way

The set S consisting of those positive integers n which are uniquely expressible in the form n = a² + b² + c², $\text{a ≧ b ≧ c ≧ 0}$, is considered. Since n ∈ S if and only if 4n ∈ S, we may restrict attention to those n not divisible by 4. Classical formulas and the theorem that there are only finitely many imaginary quadratic fields with given class number imply that there are only finitely many n ∈ S with n = 0 (mod 4). More specifically, from the existing knowledge of all the imaginary quadratic fields with odd discriminant and class number 1 or 2 it is readily deduced that there are precisely twelve positive integers n such that n ∈ S and n ≡ 3 (mod 8). To determine those n ∈ S such that n ≡ 1, 2, 5, 6 (mod 8) requires the determination of the imaginary quadratic fields with even discriminant and class number 1, 2, or 4. While the latter information is known empirically, it has not been proved that the known list of 33 such fields is complete. If it is complete, then our arguments show that there are exactly 21 positive integers n such that n ∈ S and n ≡ 1, 2, 5, 6 (mod 8).     

P4‐decompositions of regular graphs

In this article, we show that every simple r‐regular graph G admits a balanced P₄‐decomposition if r ≡ 0(mod 3) and G has no cut‐edge when r is odd. We also show that a connected 4‐regular graph G admits a P₄‐decomposition if and only if |E(G)| ≡ 0(mod 3) by characterizing graphs of maximum degree 4 that admit a triangle‐free Eulerian tour. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 135–143, 1999