We introduce a closure concept for 2-factors in claw-free graphs that generalizes the closure introduced by the first author. The 2-factor closure of a graph is uniquely determined and the closure operation turns a claw-free graph into the line graph of a graph containing no cycles of length at most 5 and no cycles of length 6 satisfying a certain condition. A graph has a 2-factor if and only i...

A gracious labelling g of a tree is a graceful labelling in which, treating the tree as a bipartite graph, the label of any edge (d,u) (d a ‘down’ and u an ‘up’ vertex) is g(u) – g(d). A gracious k-labelling is one such that each residue class modulo k has the ‘correct’ numbers of vertex and edge labels that is, the numbers that arise by interpreting the labels of a gracious labelling modulo k....

The group PGL(2, q), q = pn, p an odd prime, is 3-transitive on the projective line and therefore it can be used to construct 3-designs. In this paper, we determine the sizes of orbits from the action of PGL(2, q) on the k-subsets of the projective line when k is not congruent to 0 and 1 modulo p. Consequently, we find all values of λ for which there exist 3-(q+1, k, λ) designs admitting PGL(2,...

Let G be any of the groups (P )GL(n, q), (P )SL(n, q). Define a (simple) graph Γ = Γ(G) on the set of elements of G by connecting two vertices by an edge if and only if they generate G. We prove that if the dimension n is at least 12 and not congruent to 2 modulo 4, then the maximum size of a complete subgraph in Γ is equal to the chromatic number of Γ. This work was motivated by a question of ...

Let G(◦) and G(∗) be two groups of finite order n, and suppose that they share a normal subgroup S such that u ◦ v = u ∗ v if u ∈ S or v ∈ S. Cases when G/S is cyclic or dihedral and when u ◦ v = u ∗ v for exactly n2/4 pairs (u, v) ∈ G× G have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions...

We consider Weil sums of binomials of the form WF,d(a) = ∑ x∈F ψ(x d − ax), where F is a finite field, ψ : F → C is the canonical additive character, gcd(d, |F|) = 1, and a ∈ F. If we fix F and d and examine the values of WF,d(a) as a runs through F , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo F). Choices of F and d for whic...

An infinite group with supersimple theory has a finite series of definable groups whose factors are infinite and either virtually-FC or virtuallysimple modulo a finite FC-centre. We deduce that a group which is typedefinable in a supersimple theory has a finite series of relatively definable groups whose factors are either abelian or simple groups. In this decomposition, the non-abelian simple ...

GReTL (Graph Repository Transformation Language, [5]) is the operational transformation language of the TGraph technological space [2]. Models are represented as typed, directed, ordered, and attributed graphs. GReTL uses the GReQL (Graph Repository Query Language, [3]) for its matching part. The standard GReTL operations are currently targeted at out-place transformation which create the targe...

The set of operations in state-of-the-art graph transformation tools allows one to conditionally create and remove nodes and edges from input graphs. Node attributes can be initialized or updated with information from other attributes, parameters or constants. These operations appear to be too restricted for expressing model refinements in a concise manner. More specifically, graph transformati...