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Jan Florek

Publications and source records attributed to Jan Florek.

15 recordsLinked to original sources
arXiv2025

Kempe equivalence of 4-colourings of some plane triangulations

Let $G_{n}$, where $n \geqslant 5$, be a simple plane triangulation which has $2$ non-adjacent vertices of degree $n$ (called \textit{poles} of $G_n$) and $2n$ vertices of degree~$5$. A set of Kempe equivalent $4$-colourings of $G_{n}$ is called a \textit{Kempe class}. The number of Kempe classes of $G_{n}$ is enumerated. In particular it is shown that there is at least $\lfloor \frac{n}{6} \rfloor$ Kempe classes of $G_{n}$. We say that $4$-colourings $A, B$ of $G_{n}$ are \textit{equal} if there exists a permutation~$P$ of the set of colours such that $A = P \circ B$. Otherwise, $A$, $B$ are \textit{different}. The number of different $4$-colourings of $G_{n}$ is enumerated. Suppose that $H_{n} = G_{n} - b$, where $b$ is a pole of $G_{n}$. We prove that all $4$-colourings of $H_{n}$ are Kempe equivalent up to $\lfloor \frac{13n}{2} \rfloor$ Kempe changes. %$3n$ ($\lfloor \frac{9n}{2} \rfloor$ and $\lfloor \frac{13n}{2} \rfloor$) Kempe changes, for $n \equiv 0\, (mod\, 3)$ ($n \equiv 2\, (mod\, 3)$ and $n \equiv 1\, (mod\, 3)$, respectively).

math.CO↗
arXiv2025

The enumeration of plane (3,6)-triangulations and the form of odd perfect numbers

Let $\mathcal{P}$ be the family of all $3$-connected plane triangulations with vertices of degree $3$ or $6$. The number $d(n)$ of non-isomorphic triangulations belonging to $\mathcal{P}$ of degree $2n+2$ are enumerated, for $n \geqslant 1$. From the formula for $d(n)$ (where $n$ is odd and $n > 1$), we obtain a strengthening of Euler's theorem regarding the form of odd perfect number.

math.CO↗
arXiv2025

On Dirac and Motzkin problem in discrete geometry

Dirac and Motzkin conjectured that any set X of $n$ non-collinear points in the plane has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X. In this paper we prove that any set X of $n$ non-collinear points in the plane, distributed on three lines passing through a common point, has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X.

math.CO↗
arXiv2023

A sufficient condition for cubic 3-connected plane bipartite graphs to be hamiltonian

Barnette's conjecture asserts that every cubic $3$-connected plane bipartite graph is hamiltonian. Although, in general, the problem is still open, some partial results are known. In particular, let us call a face of a plane graph big (small) if it has at least six edges (it has four edges, respectively). Goodey proved for a $3$-connected bipartite cubic plane graph $P$, that if all big faces in $P$ have exactly six edges, then $P$ is hamiltonian. In this paper we prove that the same is true under the condition that no face in $P$ has more than four big neighbours. We also prove, that if each vertex in $P$ is incident both with a small and a big face, then~$P$ has at least $2^{k}$ different Hamilton cycles, where $k = \left\lceil\frac{|B|-2}{4\Delta(B) - 7}\right\rceil$, $|B|$ is the number of big faces in $P$ and $\Delta(B)$ is the maximum size of faces in $P$. 15 pages

math.CO↗
arXiv2023

The Stein theorem for loopless 2-connected plane multigraphs

Stein proved that for each simple plane triangulation H there exists a partitioning of the vertex of H into two subsets each of which induces a forest if and only if the dual H^{*} has a Hamilton cycle. We extend the Stein theorem for graphs in the family of all loopless 2-connected plane multigraphs and we prove some other equivalent results.

math.CO↗
arXiv2020

Graphs with multi-$4$-cycles and the Barnette's conjecture

Let ${\cal H}$ denote the family of all graphs with multi-$4$-cycles and suppose that $G \in {\cal H}$. Then, $G$ is a bipartite graph with a vertex bipartition $\{V_{\alpha}, V_{\beta}\}$. We prove that for every vertex $v \in V_{\beta}$ and for every $2$-colouring $V_{\alpha} \rightarrow \{1, 2\}$ there exists a $2$-colouring $V_{\beta} \rightarrow \{1, 2\}$ such that every cycle in $G$ is not monochromatic and $b(v) = 1$ ($b(v) = 2$). Let now $G$ be a simple even plane triangulation with a vertex $3$-partition $\{V_{1}, V_{2}, V_{3}\}$. Denote by $B_{i}$, $i = 1, 2, 3$, the set of all vertices in $V_i$ of degree at least $6$ in $G$. Suppose that $G[B_{1}\cup B_{3}]$ ($G[B_{2}\cup B_{3}]$) is a subgraph of $G$ induced by the set $B_{1}\cup B_{3}$ ($B_{2}\cup B_{3}$, respectively). Let $G^{*}$ be the dual graph of $G$ with the following $3$-face-colouring: a face $f$ of $G^{*}$ is coloured with $i$ if and only if the vertex $v = f^{*} \in V_{i}$. We prove that if $H = G[B_{1}\cup B_{3}] \cup G[B_{2}\cup B_{3}] \in {\cal H}$, then, for any edge chosen on a face coloured $3$ and of size at least $6$ in $G^{*}$, there exists a Hamilton cycle of $G^{*}$ which avoids this edge. Moreover, if every component of $H$ is $2$-connected, then there exists a Hamilton cycle of $G^{*}$ such that for every face coloured $3$ it avoids every second edge of this face or it avoids at most two edges of this face.

math.CO↗
arXiv2018

Remarks on Barnette's Conjecture

Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{*}|}{\Delta^{2}{(P^{*})}}}$ different Hamilton cycles.

math.CO↗
arXiv2016

Hamiltonian cycles in some family of cubic $3$-connected plane graphs

Barnette conjectured that all cubic $3$-connected plane graphs with maximum face size at most $6$ are hamiltonian. We provide a method of construction of a hamiltonian cycle (in dual terms) in an arbitrary cubic, $3$-connected plane graph possessing such a face $g$ that every face incident with $g$ has at most $5$ edges and every other face has at most $6$ edges.

math.CO↗
arXiv2012

On Barnette's Conjecture and $H^{+-}$ property

A conjecture of Barnette states that every 3-connected cubic bipartite plane graph has a Hamilton cycle, which is equivalent to the statement that every simple even plane triangulation admits a partition of its vertex set into two subsets so that each induces a tree. Let $G$ be a simple even plane triangulation and suppose that ${V_1, V_2, V_3}$ is a 3-coloring of the vertex set of $G$. Let $B_{i}$, $i = 1, 2, 3$, be the set of all vertices in $V_i$ of the degree at least 6. We prove that if induced graphs $G[B_1 \cup B_2]$ and $G[B_1 \cup B_3]$ are acyclic, then the following properties are satisfied: [6pt] (1) For every path $abc$ there is possible to partition the vertex set of $G$ into two subsets so that each induces a tree, and one of them contains the edge $ab$ and avoids the vertex $c$, [6pt] (2) For every path $abc$ with vertices $a$, $c$ of the same color there is possible to partition the vertex set of $G$ into two subsets so that each induces a tree, and one of them contains the path $abc$.

math.CO↗
arXiv2012

Billiards and the Five Distance Theorem II

We consider a billiard table rectangle. If a billiard ball is sent out from position F(1) at the angle of $π/4$, then the ball will rebound against the sides of the rectangle consecutively in points $F(2),F(3),...$. Let $n\geq5$ and $Φ= \{F(j): 1\leq j\leq n \}$ be the set of different points. An open connected subset of the perimeter of the billiard rectangle with different endpoints from the set $Φ$ is called \textit{segment}. \textit{Length} of a segment is a distance along the perimeter between its endpoints. A segment with endpoints $F(k)$, F(l), $1\le k,l\le n$, is called \textit{even} (or \textit{odd}), and has \textit{weight} $|k-l|$ (or $k+l$) if $k$, $l$ are of the same (or different) parity. A segment is called \textit{elementary} if there are no points of the set $Φ$ between its endpoints. Suppose $\emptyset \neq V\subseteq\{F(1),F(n)\}$. A segment $I$ is \textit{associated} with $V$ if $I$ is an elementary segment incident with an element of $V$ or $I\capΦ$ is nonempty set contained in $V$. Let $ω_1<ω_2$ be odd weights and $ω_0$ be an even weight of segments associated with $\{F(1)\}$, and let $ω_3<ω_4$ be other odd weights of segments associated with $\{F(1),F(n)\}$. Suppose that $a_i$ is the length of the segment with the weight $ω_i$, $i = 0,..., 4$. In an earlier paper the author have proved that the weights of elementary segments have at most five different values $ω_0,..., ω_4$. Moreover, elementary segments with equal weights have equal lengths. Let $A_i$ be the set of all elementary segments with weight $ω_i$. In this paper we prove that, if we know weights $ω_0$, $ω_1$, $ω_4$, and $ε, δ\in \{-1, 1\}$ such that $a_2-εa_1 = a_3-δa_4 =a_0$, then we can easily calculate $|A_0|, ..., |A_4|$.

math.CO↗
arXiv2012

Arc-Disjoint Cycles and Feedback Arc Sets

Isaak posed the following problem. Suppose $T$ is a tournament having a minimum feedback arc set which induces an acyclic digraph with a hamiltonian path. Is it true that the maximum number of arc-disjoint cycles in $T$ equals the cardinality of minimum feedback arc set of $T$? We prove that the answer to the problem is in the negative. Further, we study the number of arc-disjoint cycles through a vertex $v$ of the minimum out-degree in an oriented graph $D$. We prove that if $v$ is adjacent to all other vertices, then $v$ belongs to $δ^+(D)$ arc-disjoint cycles.

math.CO↗
arXiv2012

Orbits and Hamilton bonds in a family of plane triangulations with vertices of degree three or six

Let $\cal{P}$ be the family of all 2-connected plane triangulations with vertices of degree three or six. Grünbaum and Motzkin proved (in the dual terms) that every graph $P \in \cal{P}$ is factorable into factors $P_0$, $P_1$, $P_2$ (indexed by elements of the cyclic group $Q = \{0,1,2\}$) such that every factor $P_q$ consists of two induced paths with the same length $M(q)$, and $K(q)-1$ induced cycles with the same length $2M(q)$. For $q \in Q$, we define an integer $S^+(q)$ such that the vector $(K(q), M(q), S^+(q))$ determines the graph $P$ (if $P$ is simple) uniquely up to orientation-preserving isomorphism. We establish arithmetic equations that will allow calculate the vector $(K(q+1), M(q+1), S^+(q+1))$ by the vector $(K(q), M(q), S^+(q))$, $q \in Q$. We present some applications of the equations. The set $\{(K(q), M(q), S^+(q)): q \in Q\}$ is called the orbit of $P$. We characterize one point orbits of graphs in $\cal{P}$. We prove that if $P$ is of order $4n +2$, $n \in\mathbb{N}$, than it has a Hamilton bond such that the end-trees of the bond are equitable 2-colorable and have the same order. We prove that if $M(q)$ is odd and $K(q) \geqslant \frac{M(q)}{3}$, then there are two disjoint induced paths of the same order, which vertices together span all of $P$.

math.CO↗
arXiv2011

Roots of Markoff quadratic forms as strongly badly approximable numbers

For a real number $x$, $\| x\| = \min \{|x-p|: p\in Z\}$ is the distance of $x$ to the nearest integer. We say that two real numbers $θ$, $θ'$ are $\pm$ equivalent if their sum or difference is an integer. Let $θ$ be irrational and put \[ϕ(θ) = \inf \{q \,\| q θ\| : q \in N \}. \] We will prove: If $ϕ(θ)> 1/3$, then $θ$ is $\pm$ equivalent to a root of $f_m (x,1) = 0$, where $f_m$ is a Markoff form. Conversely, if $θ$ is $\pm$ equivalent to a root of $f_m(x,1)=0$, then \[ϕ(θ) = m \| mθ\| = \frac{2}{3+\sqrt{9-4m^{-2}}} > 1/3. \]

math.NT↗
arXiv2011

Allocation of seats in the European Parliament and a degressive proportionality

Distribution of seats in The European Parliament postulated by Treaty of Lisbon should be degressively proportional. The meaning of degressively proportional concept can be found in two principles annexed to the draft of European Parliament resolution. The first, referred as the principle of fair division, states that "the larger the population of a Member State, the greater is entitlement to a large number of seats". The other condition, referred to as the principle of relative proportionality, holds that "the larger the population of a country, the more inhabitants are represented by each of its Members of the EU". We postulate a clear and fair method which determines uniquely a distribution of seats in the European Parliament which fulfil the requirements of degressive proportionality. More generally, let $l_i$ be any non-increasing sequence of real positive numbers. We say that a sequence of natural numbers $m_{i}$ is degressively proportional with respect to the sequence $l_{i}$, if $m_{i}$ and $l_{i}/m_{i}$ are non-increasing sequences. Our method can be instrumental in uniquely determining a degressively proportional sequence $m_{i}$ with respect to $l_{i}$ which fulfils given conditions.

physics.soc-ph↗