Dominating sets in projective planes

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result which shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order $q>81$ is smaller than $2q+2[\sqrt{q}]+2$ (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most $2q+\sqrt{q}+1$. In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.Comment: 19 page

Search Problems in Vector Spaces

We consider the following $q$-analog of the basic combinatorial search problem: let $q$ be a prime power and \GF(q) the finite field of $q$ elements. Let $V$ denote an $n$-dimensional vector space over \GF(q) and let $\mathbf{v}$ be an unknown 1-dimensional subspace of $V$. We will be interested in determining the minimum number of queries that is needed to find $\mathbf{v}$ provided all queries are subspaces of $V$ and the answer to a query $U$ is YES if $\mathbf{v} \leqslant U$ and NO if $\mathbf{v} \not\leqslant U$. This number will be denoted by $A(n,q)$ in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and $M(n,q)$ in the non-adaptive case (when all queries must be made in advance). In the case $n=3$ we prove $2q-1=A(3,q)<M(3,q)$ if $q$ is large enough. While for general values of $n$ and $q$ we establish the bounds $$n\log q \le A(n,q) \le (1+o(1))nq$$ and $$(1-o(1))nq \le M(n,q) \le 2nq,$$ provided $q$ tends to infinity

On geometric constructions of (k, g)-graphs

We give new constructions for k-regular graphs of girth 6, 8 and 12 with a small number of vertices. The key idea is to start with a generalized n-gon and delete some lines and points to decrease the valency of the incidence graph