Diagrams that serve as a visual input facility for programming environments have to be translated into some kind of semantic description. This paper describes such a method which is based on a specification of the translation process. The translation process starts with a diagram, which is simply represented as a collection of atomic diagram components, and it ends up with some data structure a...

Let S be a set of transpositions that generates the symmetric group Sn, where n ≥ 3. The transposition graph T (S) is defined to be the graph with vertex set {1, . . . , n} and with vertices i and j being adjacent in T (S) whenever (i, j) ∈ S. We prove that if the girth of the transposition graph T (S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the semidirect p...

Let Γ be a Cayley graph of the permutation group generated by a transposition tree T on n vertices. In an oft-cited paper [2] (see also [14]), it is shown that the diameter of the Cayley graph Γ is bounded as diam(Γ) ≤ max π∈Sn {

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).

For a tree T , the graph X is T -decomposable if there exists a partition of the edge set of X into isomorphic copies of T . In 1963, Ringel conjectured that K2m+1 can be decomposed by any tree with m edges. Graham and Häggkvist conjectured more generally that every 2m-regular graph can be decomposed by any tree with m edges. Fink showed in 1994 that for any directed tree T with m arcs, the dir...

A mixed hypergraph consists of two families of edges: the C-edges and D-edges. In a coloring every C-edge has at least two vertices of the same color, while every D-edge has at least two vertices colored differently. The largest and smallest possible numbers of colors in a coloring are termed the upper and lower chromatic number, χ̄ and χ, respectively. A mixed hypergraph is called uniquely colo...

We consider a discrete-time quantum walk Wt given by the Grover transformation on the Cayley tree. We reduce Wt to a quantum walk Xt on a half line with a wall at the origin. This paper presents two types of limit theorems for Xt. The first one is Xt as t → ∞, which corresponds to a localization in the case of an initial qubit state. The second one is Xt/t as t → ∞, whose limit density is given...

Let H = (V,E) be an r-uniform hypergraph and let F ⊂ 2 . A matching M of H is (α,F)perfect if for each F ∈ F , at least α|F | vertices of F are covered by M . Our main result is a theorem giving sufficient conditions for an r-uniform hypergraph to have a (1− ,F)-perfect matching. As a special case of our theorem we obtain the following result. Let K(n, r) denote the complete r-uniform hypergrap...

A characterisation is given of edge-transitive Cayley graphs of valency 4 on odd number of vertices. The characterisation is then applied to solve several problems in the area of edge-transitive graphs: answering a question proposed by Xu (1998) regarding normal Cayley graphs; providing a method for constructing edge-transitive graphs of valency 4 with arbitrarily large vertex-stabiliser; const...

For two normal edge-transitive Cayley graphs on groups H and K which have no common direct factor and gcd(|H/H ′|, |Z(K)|) = 1 = gcd(|K/K′|, |Z(H)|), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive. c ⃝ 2014 IAUCTB. All rights reserved.