A tournament is an oriented complete graph, and one containing no directed cycles is called transitive. A tournament T= (V,A) is called m-partition transitive if there is a partition V=X1∪· X2∪· · · ·∪· Xm such that the subtournaments induced by each Xi are all transitive, and T Contract grant sponsor: University of Dayton Research Council (to A. H. B.); Contract grant sponsor: National Science...

Two questions are considered, namely (i) How many colors are needed for a coloring of the n-cube without monochromatic quadrangles or hexagons? We show that four colors suffice and thereby settle a problem of Erdos. (ii) Which vertex-transitive induced subgraphs does a hypercube have? An interesting graph has come up in this context: If we delete a Hamming code from the 7-cube, the resulting gr...

A permutation group is said to be quasiprimitive if each of its nontrivial normal subgroups is transitive. A structure theorem for finite quasiprimitive permutation groups is proved, along the lines of the O'NanScott Theorem for finite primitive permutation groups. It is shown that every finite, non-bipartite, 2-arc transitive graph is a cover of a quasiprimitive 2-arc transitive graph. The str...

A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a directi...

A transitive orientation of a graph is an orientation of the edges that produces a transitive digraph. The modular decomposition of a graph is a canonical representation of all of its modules. Finding a transitive orientation and finding the modular decomposition are in some sense dual problems. In this paper, we describe a simple O(n+mlogn) algorithm that uses this duality to find both a trans...

We consider a code to be a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of vertices, not in the code, at distance s from some codeword, but not distance less than s from any codeword. A 2-neighbour transitive code is a code which admits a group X of automorphisms which is transitive on the s-neighbours, for s = 1, 2, and transitive on the code itself...

Let ? be a finite, undirected, connected, simple graph. We say that matching M is permutable m -matching if contains edges and the subgroup of Aut ( ) fixes setwise allows to permuted in any fashion. A 2-transitive stabilizer can map ordered pair distinct other . provide constructions graphs with matching; we show that, an arc-transitive graph for ? 4 , then degree at least ; and, when sufficie...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a graph G is defined to have vertices the arcs of G such that two arcs uv, xy are adjacent if and only if (v, u, x, y) is a 3-arc of G. In this paper we prove that any connected 3-arc graph is Hamiltonian, and all iterative 3-...

‎let $g=(v(g),e(g))$ be a simple connected graph with vertex set $v(g)$ and edge‎ ‎set $e(g)$‎. ‎the (first) edge-hyper wiener index of the graph $g$ is defined as‎: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=s...

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs admitting an automorphism groupG that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph B corresponding toB. If in additionG is ...