Related papers: Minimal complete arcs in PG(2,q), q<= 32

Let ind-match$(G)$, min-match$(G)$ and match$(G)$ denote the induced matching number, minimum matching number and matching number of a graph $G$, respectively. It is known that ind-match$(G) \leq $ min-match$(G) \leq {\rm match}(G) \leq$…

We show that the metric dimension of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in $\mathrm{PG}(2,q)$, the…

Minimal 1-saturating sets in the projective plane $PG(2,q)$ are considered. They correspond to covering codes which can be applied to many branches of combinatorics and information theory, as data compression, compression with distortion,…

In this work complete caps in $PG(N,q)$ of size $O(q^{\frac{N-1}{2}}\log^{300} q)$ are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $\sqrt{2}q^{\frac{N-1}{2}}$ and it…

The classification of all semiovals and blocking semiovals in $\mathrm{PG}(2,8)$ and in $\mathrm{PG}(2,9)$ of size less than $17$ is determined. Also, some information on the stabilizer groups and the intersection sizes with lines is given.

In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a…

Small complete arcs and caps in Galois spaces over finite fields $\fq$ with characteristic greater than 3 are constructed from cubic curves with an isolated double point. For $m$ a divisor of $q+1$, complete plane arcs of size approximately…

A proper circular-arc (PCA) model is a pair ${\cal M} = (C, \cal A)$ where $C$ is a circle and $\cal A$ is a family of inclusion-free arcs on $C$ in which no two arcs of $\cal A$ cover $C$. A PCA model $\cal U = (C,\cal A)$ is a $(c,…

The identity of the smallest quadrangulation with minimum degree 3 also containing parallel edges is unknown. However, it has already been determined that its order (the number of vertices) is between 11 and 14. This paper narrows this…

In a projective plane over a finite field, complete $(k,n)$-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the…

We prove that, if $q$ is large enough, the set of the $\mathbb{F}_{q^6}$-rational points of the Hermitian curve is a complete $(q+1)$-arc in $\mathrm{PG}(2,\mathbb{F}_{q^6})$, addressing an open case from a recent paper by Korchm\'aros,…

Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with…

Let $G$ be a graph with a perfect matching. Denote by $f(G)$ the minimum size of a matching in $G$ which is uniquely extendable to a perfect matching in $G$. Diwan (2019) proved by linear algebra that for $d$-hypercube $Q_d$ ($d\geq 2)$,…

The components of the graphs $D(n, q)$ provide the best-known general lower bound for the number of edges in a graph with $n$ vertices and no cycles of length less than $g$. In this paper, we give a new, short, and simpler proof of the fact…

We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine…

An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.

In this work, we prove the existence of maximal partial line spreads in PG(5,q) of size q^3+q^2+kq+1, with 1 \leq k \leq (q^3-q^2)/(q+1), k an integer. Moreover, by a computer search, we do this for larger values of k, for q \leq 7. Again…

A subset $\mathcal{S}$ of a conic $\mathcal{C}$ in the projective plane $\mathrm{PG}(2,q)$ is called almost complete (AC-subset for short) if it can be extended to a larger arc in $\mathrm{PG}(2,q)$ only by the points of…

We describe an algorithm for testing the completeness of caps in PG(r; q), q even. It allowed us to check that the 95256-cap in PG(12; 4) recently found by Fu el al. (see [14]) is complete.

This paper describes an efficient approach to constructing a resultant polyline with a minimum number of segments and arcs. While fitting an arc can be done with complexity O(1) (see [1] and [2]), the main complexity is in checking that the…