We resolve two problems of [Cameron, Praeger, and Wormald – Infinite highly arc transitive digraphs and universal covering digraphs, Combinatorica 1993]. First, we construct a locally finite highly arc-transitive digraph with universal reachability relation. Second, we provide constructions of 2-ended highly arc transitive digraphs where each ‘building block’ is a finite bipartite graph that is...

A circulant is a Cayley digraph over a finite cyclic group. The classification of arc-transitive circulants is shown. The result follows from earlier descriptions of Schur rings over cyclic groups.

We show that a necessary and sufficient condition for the existence of a Kp,q factorization of the symmetric complete bipartite multi-digraph )"K:n n is (i) m = n == 0 (mod p) for p = q and (ii) m = n == a (mod d(p' + q')p'q'/e) for p =f. q, where d = (p,q), p' = p/d, q' = q/d, e = ()..,p'q').

We show that necessary and sufficient conditions for the existence of a semi-evenly partite star factorization of the symmetric complete tripartite digraph K~I,n2,n3 are (i) k is even, k 2 4 and (ii) nl = n2 = n3 == 0 (mod k(k -1)/3) for k == 0 (mod 6) and nl = n2 = n3 == 0 (mod k(k 1)) for k == 2,4 (mod 6).

Let {Gp1, Gp2, . . .} be an infinite sequence of graphs with Gpn having pn vertices. This sequence is called Kp-removable if Gp1 ∼= Kp, and Gpn − S ∼= Gp(n−1) for every n ≥ 2 and every vertex subset S of Gpn that induces a Kp. Each graph in such a sequence has a high degree of symmetry: every way of removing the vertices of any fixed number of disjoint Kp’s yields the same subgraph. Here we con...

4 We prove that the weak k-linkage problem is polynomial for every fixed k for totally Φ5 decomposable digraphs, under appropriate hypothesis on Φ. We then apply this and recent results 6 by Fradkin and Seymour (on the weak k-linkage problem for digraphs of bounded independence 7 number or bounded cut-width) to get polynomial algorithms for some class of digraphs like quasi8 transitive digraphs...

$D$ if $D^{j}$ is a g-cyclic A-cover of $D$ for some $g\in A$ . In the case that $A$ is abelian, . then $D_{g}(\alpha)$ is simply called a cyclic abelian cover. Let $\alpha$ and $\beta$ be two alternating functions $homA(D)$ into $A$ , and let $\Gamma$ be a subgroup of the automorphism group Aut $D$ of $D$ , denoted $\Gamma\leq$ Aut $D$ . Let $g,$ $h\in A$ . Then two cyclic A-covers $D_{g}(\alp...

We show that a necessary and sufficient condition for the existence of a Kp ,2q factorization of the symmetric complete tripartite digraph K~1,n2,n3 is (i) ni = n2 = n3 == 0 (mod p) for p = q, (ii) ni = n2 = n3 == 0 (mod dp'q'(p' + 2q')) for p =Iq and p' odd, (iii) ni = n2 = n3 == 0 (mod dp'q'(p' + 2q')/2) for p =Iq and p' even, where d = (p, q), p' = p/d, q' = q/d.

We obtain upper bounds for the upper generalized exponents of digraphs in the class of ministrong digraphs and in the class of non-primitive ministrong digraphs, characterize the corresponding extremal digraphs, and discuss the numbers attainable as upper generalized exponents of ministrong digraphs.