On Some Cycles in Wenger Graphs

Let p be a prime, q be a power of p, and let F q be the field of q elements. For any positive integer n, the Wenger graph W n (q) is defined as follows: it is a bipartite graph with the vertex partitions being two copies of the (n + 1)-dimensional vector space F n+1 , and two vertices p = (p(1), . . . , p(n + 1)), and l = [l(1), . . . , l(n + 1)] q being adjacent if p(i) + l(i) = p(1)l(1) i−1 , for all i = 2, 3, . . . , n + 1

Cycles, wheels, and gears in finite planes

The existence of a primitive element of $GF(q)$ with certain properties is used to prove that all cycles that could theoretically be embedded in $AG(2,q)$ and $PG(2,q)$ can, in fact, be embedded there (i.e. these planes are `pancyclic'). We also study embeddings of wheel and gear graphs in arbitrary projective planes

On graphs with unique geoodesics and antipodes

In 1962, Oystein Ore asked in which graphs there is exactly one geodesic between any two vertices. He called such graphs geodetic. In this paper, we systematically study properties of geodetic graphs, and also consider antipodal graphs, in which each vertex has exactly one antipode (a farthest vertex). We find necessary and sufficient conditions for a graph to be geodetic or antipodal, obtain results related to algorithmic construction, and find interesting families of Hamiltonian geodetic graphs. By introducing and describing the maximal hereditary subclasses and the minimal hereditary superclasses of the geodetic and antipodal graphs, we get close to the goal of our research -- a constructive classification of these graphs

EMBEDDING CYCLES IN FINITE PLANES

Abstract. We define and study embeddings of cycles in finite affine and projective planes. We show that for all k, 3 ≤ k ≤ q 2, a k-cycle can be embedded in any affine plane of order q. We also prove a similar result for finite projective planes: for all k, 3 ≤ k ≤ q 2 + q + 1, a k-cycle can be embedded in any projective plane of order q. 1