arXiv subjects

Vladislav Taranchuk

Publications and source records attributed to Vladislav Taranchuk.

At least 19 recordsLinked to original sources

arXiv2026

Recovery Models for Linear Batch Codes

Various types of recovery algorithms for batch codes have been considered previously, including asynchronous recovery and recovery algorithms arising from Almost Affinely Disjoint (AAD) families. We initiate a systematic investigation of linear batch codes equipped with recovery algorithms. We introduce online and strongly online batch codes, investigate their relations with asynchronous batch codes, and introduce (m,L)-strongly online batch codes, which provide particularly simple recovery algorithms. We then study simple batch codes associated with arbitrary bipartite graphs, obtaining graph-theoretic conditions for such codes and generalizing several known results.

cs.IT↗

arXiv2026

A note on the chromatic number of Kneser graphs on chambers of projective planes and incidence-free sets

Let $D=(\mathcal{P},\mathcal{B})$ be a symmetric $(v,k,λ)$-design and let $(X,Y)$ be an equinumerous incidence-free pair, with $X\subseteq \mathcal{P}$ and $Y\subseteq \mathcal{B}$. In this note, we give an elementary proof which shows the existence of a perfect matching between $\mathcal{P} \setminus X$ and $\mathcal{B}\setminus Y$ in the incidence graph of $D$. This recovers a result of Spiro, Adriaensen and Mattheus, who already showed this using different arguments for $k\geq 36$. We use this to connect some dots in the literature and prove that finding the chromatic number of the Kneser graph on chambers of a projective plane is equivalent to finding the incidence-free number of the incidence graph of the plane. Furthermore, we construct an incidence-free pair for PG$(2,q^2)$ of size roughly $q^3/2+3q^2/4$.

math.CO↗

arXiv2026

$K_{2, t+1}$-free graphs containing an optimal number of $K_{t, t}$'s

The generalized Turán number $ex(n, K_{t, t}, K_{2, t+1})$ is the maximum number of copies of $K_{t, t}$ that a $K_{2, t+1}$-free graph on $n$ vertices can contain. Recently, Pohoata, Tidor, and Yu established that $ex(n, K_{t, t}, K_{2, t+1}) = Θ_t(n^2)$ for all integers $t \geq 3$. In this short note, we use an explicit construction to establish that when $t$ is a prime power and $n = t^{2e - 1}$, then $$ ex(n, K_{t, t}, K_{2, t+1}) = (1 + o(1))\frac{n^2}{2t(t-1)}. $$

math.CO↗

arXiv2026

On Reed-Muller subcodes, Grassmannian partitions and sum-free functions

A function $F:\mathbb{F}_{2}^{n}\to \mathbb{F}_{2}^{m}$ is called $k$th-order sum-free if the sum of its values over any $k$-dimensional affine subspace of $\mathbb{F}_2^n$ is non-zero. Carlet recently introduced this notion and constructed such functions for every $2\le k\le n$. We prove that, for $2\le k\le n-2$ and $m \leq n$, the existence of a (non-degenerate) $\mathbb{F}_{2}^{m}$-valued $k$th-order sum-free function on $\mathbb{F}_{2}^{n}$ is equivalent to the existence of a codimension $m$ linear subcode of the Reed-Muller code $\mathrm{RM}(n-k,n)$ with minimum distance $3\cdot 2^{k-1}$. In particular, this yields a new family of Reed-Muller subcodes that avoid all minimum weight codewords of $\mathrm{RM}(n-k,n)$, and thus have minimum distance $3/2$ times that of $\mathrm{RM}(n-k,n)$. We also derive new necessary conditions for the existence of $k$th-order sum-free functions and present the first nontrivial lower bound on $m$. Finally, we observe that $k$th-order sum-free functions lead to a partition of the Grassmannian of all $k$-dimensional (linear) subspaces of $\mathbb{F}_2^n$ into constant-dimension subspace codes. Under the assumption that functions exist that are $k$th-order sum-free for multiple values of $k$, we obtain an improved partitioning result and a stronger upper bound on the chromatic number of the Grassmann graphs.

cs.IT↗

arXiv2026

Chromatic Number of Grassmann Graphs and MRD codes

In this paper we investigate the chromatic number of the Grassmann graphs and of their powers, denoted $J_q(n,m,t)$. In this graph, the vertices correspond to the $m$-dimensional subspaces in $\mathbb{F}_q^n$ and two vertices are adjacent if the corresponding subspaces intersect in a subspace of dimension at least $t$. By generalizing the lifting technique of Silva, Kötter and Kschischang, we use \emph{maximum rank distance (MRD)} codes to establish that $χ(J_q(n, m, t)) \leq (1 +o(1))n^{m-t}q^{(n-m)(m-t)})$ when $n \geq 2m$. Given that $J_q(n, m, t)$ is isomorphic to $J_q(n,n-m,n-2m+t)$, this establishes a new upper bound on $J_q(n, m, t)$ for any valid choice of parameters. Furthermore, we observe that in the regime that $n, m $, and $t$ are fixed, our bound is asymptotically tight, implying that $ χ(J_q(n, m, t)) = Θ(q^{(m-t)\max(n-m, m)}). $

math.CO↗

arXiv2026

Large line-free sets and their applications

In this paper, we construct explicit families of polynomials $P \in \mathbb{F}_q[x_1,\dots,x_n]$ with large root sets which have restricted intersections with affine lines. We use these sets to make substantial progress on a number of problems in extremal combinatorics. For each prime power $q$ and integer $2 \le t \le q-1$, we construct $t$-line evasive subsets of $\mathbb{F}_q^n$ of size \[ q^{\,n\left(1-\frac{2}{t^2+t}\right)}, \] which is significantly larger than those previously known. Moreover, our method yields a partition of $\mathbb{F}_q^n$ into such sets. We extend this partitioning result to the projective space $PG(n,q)$, obtaining the first explicit colorings for the vector space Ramsey number $R_q(2;k)$ that exhibit dependence on both $q$ and $k$. In particular, we show that \[ R_q(2;k) > \frac{(q-1)k}{2} - O_q(1), \] improving recent bounds. Finally, we apply these constructions to extremal graph theory and improve the best-known bounds on the bipartite Turán number $ \mathrm{ex}(n,m,\{C_4,θ_{3,t}\})$. Most notably, we show that \[ \mathrm{ex}(n,n^{2/3},\{C_4,θ_{3,3}\}) = Θ(n^{1+1/9}), \] making progress on a question originally posed by Erdős.

math.CO↗

arXiv2025

Examples of diameter-2 graphs with no triangle or $K_{2,t}$

For each $t \ge 1$ let $W_t$ denote the class of graphs other than stars that have diameter $2$ and contain neither a triangle nor a $K_{2,t}$. The famous Hoffman--Singleton Theorem implies that $W_2$ is finite. Recently Wood suggested the study of $W_t$ for $t > 2$ and conjectured that $W_t$ is finite for all $t \ge 2$. In this note we show that (1) $W_3$ is infinite, (2) $W_5$ contains infinitely many regular graphs, and (3) $W_7$ contains infinitely many Cayley graphs. Our $W_3$ and $W_5$ examples are based on so-called crooked graphs, first constructed by de Caen, Mathon, and Moorhouse. Our $W_7$ examples are Cayley graphs with vertex set $\mathbb{F}_p^2$ for prime $p \equiv 11 \pmod {12}$.

math.CO↗

arXiv2025

Line-parallelisms of PG$(n, 2)$ from Preparata-like codes

Partitions of the binary linear Hamming code into Preparata-like codes are known to induce line-parallelisms of PG$(n, 2)$. In this paper, we show that if $P$ is any Preparata-like code contained in the binary linear Hamming code $H$ of the same length, then $H$ can be partitioned into additive translates of $P$. This generalizes a result of Baker, van Lint, and Wilson who prove this fact for the class of generalized Preparata codes. We give an explicit description for line-parallelisms obtained from such a partition via crooked Preparata-like codes and establish an equivalence criterion for such line-parallelisms.

math.CO↗

arXiv2025

On the sum of the largest and smallest eigenvalues of odd-cycle free graphs

Let $G$ be a graph with adjacency eigenvalues $λ_1 \geq \cdots \geq λ_n$. Both $λ_1 + λ_n$ and the odd girth of $G$ can be seen as measures of the bipartiteness of $G$. Csikvári proved in 2022 that for odd girth 5 graphs (triangle-free) it holds that $(λ_1+λ_n)/n \le (3-2\sqrt 2) < 0.1716$. In this paper we extend Csikvári's result to general odd girth $k$ proving that $(λ_1+λ_n)/n = O(k^{-1})$. In the case of odd girth 7, we prove a stronger upper bound of $(λ_1+λ_n)/n < 0.0396$.

math.CO↗

arXiv2025

On the Chromatic Number of Grassmann Graphs

In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq χ(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $χ(J_q(n, 2))$ is equivalent to determining the smallest number of partial line parallelisms that one can partition the lines of PG$(n-1, q)$ into. We survey known results about line parallelisms and their implications for $χ(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $χ(J_q(n, 2)) < 2\binom{n-1}{1}_q$.

math.CO↗

arXiv2025

New constructions of unbalanced $\{C_4,θ_{3, t}\}$-free bipartite graphs

In 1979, Erdős conjectured that if $m = O(n^{2/3})$, then $ex(n, m, \{C_4, C_6 \}) = O(n)$. This conjecture was disproven by several papers and the current best-known bounds for this problem are $$ c_1n^{1 + \frac{1}{15}} \leq ex(n, n^{2/3}, \{C_4, C_6\}) \leq c_2n^{1 + 1/9} $$ for some constants $c_1, c_2$. A consequence of our work here proves that $$ ex(n, n^{2/3}, \{ C_4, θ_{3, 4} \}) = Θ(n^{1 + 1/9}). $$ More generally, for each integer $t \geq 2$, we establish that $$ ex(n, n^{\frac{t+2}{2t+1}}, \{ C_4, θ_{3, t} \}) = Θ(n^{1 + \frac{1}{2t+1}}) $$ by demonstrating that subsets of points $S \subseteq \text{PG}(n,q)$ for which no $t+1$ points lie on a line give rise to $\{ C_4, θ_{3, t} \}$-free graphs, where PG$(n,q)$ is the projective space of dimension $n$ over the finite field of $q$ elements.

math.CO↗

arXiv2024

Achromatic colorings of polarity graphs

A complete partition of a graph $G$ is a partition of the vertex set such that there is at least one edge between any two parts. The largest $r$ such that $G$ has a complete partition into $r$ parts, each of which is an independent set, is the achromatic number of $G$. We determine the achromatic number of polarity graphs of biaffine planes coming from generalized polygons. Our colorings of a family of unitary polarity graphs are used to solve a problem of Axenovich and Martin on complete partitions of $C_4$-free graphs. Furthermore, these colorings prove that there are sequences of graphs which are optimally complete and have unbounded degree, a problem that had been studied for the sequence of hypercubes independently by Roichman, and Ahlswede, Bezrukov, Blokhuis, Metsch, and Moorhouse.

math.CO↗

arXiv2024

A new lower bound for the multicolor Ramsey number $r_k(K_{2, t + 1})$

In this short note, we provide a new infinite family of $K_{2, t+1}$-free graphs for each prime power $t$. Using these graphs, we show that it is possible to partition the edges of $K_n$ into parts, such that each part is isomorphic to our $K_{2, t+1}$-free graph. This yields an improved lower bound to the multicolor Ramsey number $r_k(K_{2, t+1})$ when $k$ and $t$ are powers of the same prime. For these values of $k$ and $t$, our coloring implies that $$ tk^2 + 1 \leq r_k(K_{2, t+1}) \leq tk^2 + k + 2. $$ where the upper bound is due to Chung and Graham.

math.CO↗

arXiv2022

On the eigenvalues of the graphs $D(5, q)$

Let $q = p^e$, where $p$ is a prime and $e$ is a positive integer. The family of graphs $D(k, q)$, defined for any positive integer $k$ and prime power $q$, were introduced by Lazebnik and Ustimenko in 1995. To this day, the connected components of the graphs $D(k, q)$, provide the best known general lower bound for the size of a graph of given order and given girth. Furthermore, Ustimenko conjectured that the second largest eigenvalue of $D(k, q)$ is always less than or equal to $2\sqrt{q}$. If true, this would imply that for a fixed $q$ and $k$ growing, $D(k, q)$ would define a family of expanders that are nearly Ramanujan. In this paper we prove the smallest open case of the conjecture, showing that for all odd prime powers $q$, the second largest eigenvalue of $D(5, q)$ is less than or equal to $2\sqrt{q}$.

math.CO↗

arXiv2022

A simple proof for the lower bound of the girth of graphs $D(n, q)$

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 that the length of the shortest cycle appearing in $D(n, q)$ is $n + 5$ when $n$ is odd, and $n + 4$ when $n$ is even.

math.CO↗

arXiv2021

A New Family of Algebraically Defined Graphs With Small Automorphism Group

Let $p$ be an odd prime, $q=p^e$, $e\ge 1$, and $\mathbb{F} = \mathbb{F_q}$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of the 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite graph $Γ_\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ and every $[l]= [l_1,l_2,l_3]\in L$, $\{(p), [l]\} = (p)[l]$ is an edge in $Γ_\mathbb{F} (f, g)$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$ Given $Γ_\mathbb{F} (f, g)$, is it always possible to find a function $h:\mathbb{F}^2\to \mathbb{F}$ such that the graph $Γ_\mathbb{F} (f, h)$ with the same vertex set as $Γ_\mathbb{F} (f, g)$ and with edges $(p)[l]$ defined in a similar way by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $Γ_\mathbb{F} (f, g)$ for infinitely many $q$? In this paper we show that the answer to the question is negative and the graphs $Γ_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide such an example for $p \equiv 1 \pmod{3}$. Our argument is based on proving that the automorphism group of these graphs has order $p$, which is the smallest possible order of the automorphism group of graphs of the form $Γ_{\mathbb{F}}(f, g)$.

math.CO↗

arXiv2021

On the number of $k$-gons in finite projective planes

Let $Π$ be a projective plane of order $n$ and $Γ_Π$ be its Levi graph (the point-line incidence graph). For fixed $k \geq 3$, let $c_{2k}(Γ_Π)$ denote the number of $2k$-cycles in $Γ_Π$. In this paper we show that $$ c_{2k}(Γ_Π) = \frac{1}{2k}n^{2k} + O(n^{2k-2}), \hspace{0.5cm} n \rightarrow \infty. $$ We also state a conjecture regarding the third and fourth largest terms in the asymptotic of the number of $2k$-cycles in $Γ_Π$. This result was also obtained independently by Voropaev in 2012. Let $\text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\})$ denote the greatest number of $2k$-cycles amongst all bipartite graphs of order $v$ and girth at least 6. As a corollary of the result above, we obtain $$ \text{ex}(v, C_{2k}, \mathcal{C}_{\text{odd}}\cup \{C_4\}) = \left(\frac{1}{2^{k+1}k}-o(1)\right)v^k, \hspace{0.5cm} v \rightarrow \infty. $$

math.CO↗

arXiv2018

The Anti-Ramsey Problem for the Sidon equation

For $n \geq k \geq 4$, let $AR_{X + Y = Z + T}^k (n)$ be the maximum number of rainbow solutions to the Sidon equation $X+Y = Z + T$ over all $k$-colorings $c:[n] \rightarrow [k]$. It can be shown that the total number of solutions in $[n]$ to the Sidon equation is $n^3/12 + O(n^2)$ and so, trivially, $AR_{X+Y = Z + T}^k (n) \leq n^3 /12 + O (n^2)$. We improve this upper bound to \[ AR_{X+Y = Z+ T}^k (n) \leq \left( \frac{1}{12} - \frac{1}{24k} \right)n^3 + O_k(n^2) \] for all $n \geq k \geq 4$. Furthermore, we give an explicit $k$-coloring of $[n]$ with more rainbow solutions to the Sidon equation than a random $k$-coloring, and gives a lower bound of \[ \left( \frac{1}{12} - \frac{1}{3k} \right)n^3 - O_k (n^2) \leq AR_{X+Y = Z+ T}^k (n). \] When $k = 4$, we use a different approach based on additive energy to obtain an upper bound of $3n^3 / 96 + O(n^2)$, whereas our lower bound is $2n^3 / 96 - O (n^2)$ in this case.

math.CO↗