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Jan Ekstein

Publications and source records attributed to Jan Ekstein.

13 recordsLinked to original sources
arXiv2025

Hamiltonian paths in iterated line graphs

For integer $n$, the $n$-iterated line graph $L^n(G)$ of an undirected graph $G$ is defined to be $L(L^{n-1}(G))$, where $L^1(G)$ is the line graph $L(G)$ of $G$. In this paper we introduce hamiltonian path index. Hamiltonian path index, denoted by $h_p(G)$, is the minimum number $n$ such that $L^n(G)$ contains a hamiltonian path. We show that hamiltonian path index of $G$ exists for any graph $G$ and we set the exact value of hamiltonian path index for trees and discuss the problem about graphs with hamiltonian 2-connected blocks.

math.CO↗
arXiv2020

Hamiltonian properties in generalized lexicographic products

The lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from $G$ by replacing each vertex with a copy of $H$ and adding all edges between any pair of copies corresponding to adjacent vertices of $G$. We consider also the generalized lexicographic product such that we replace each vertex of $G$ with arbitrary graph on the same number of vertices. We present sufficient and necessary conditions for traceability, hamiltonicity and hamiltonian connectivity of $G[H]$ if $G$ is a path and hence we improved and extended results in M. Kriesell, A Note on Hamiltonian Cycles in Lexicographical Products.

math.CO↗
arXiv2018

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any 2 longest paths have a vertex in common. For k>=7, Skupien in [7] obtained a connected graph in which some k longest paths have no common vertex, but every k-1 longest paths have a common vertex. It is not known whether every 3 longest paths in a connected graph have a common vertex and similarly for 4, 5, and 6 longest path. In [5] the authors give an upper bound on distance among 3 longest paths in a connected graph. In this paper we give a similar upper bound on distance between 4 longest paths and also for k longest paths, in general.

math.CO↗
arXiv2017

Connected even factors in the square of essentially 2-edge connected graphs

In this paper we prove that the square of an essentially 2-edge connected graph with an additional property has a connected even factor with maximum degree at most 4. Moreover we show that, in general, the square of essentially 2-edge connected graph does not contain a connected even factor with bounded maximum degree.

math.CO↗
arXiv2017

Revisiting the Hamiltonian Theme in the Square of a Block: The Case of DT-Graphs

The square of a graph G, denoted G^2, is the graph obtained from G by joining by an edge any two nonadjacent vertices which have a common neighbor. A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2, ..., x_k} in G, there is a hamiltonian path from x_1 to x_2 in G^2 containing k-2 distinct edges of G of the form x_iz_i, i = 3, ..., k. It was proved many years ago that every 2-connected graph has the F_3 property. In the first part of this work, we extend this result by proving that every 2-connected DT-graph has the F_4 property (Theorem 2) and will show in the second part that this generalization holds for arbitrary 2-connected graphs, and that there exist 2-connected graphs which do not have the F_k property for any natural number k >= 5. Altogether, this answers a problem raised before in the affirmative.

math.CO↗
arXiv2013

The Packing Coloring of Distance Graphs $D(k,t)$

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than $i$. For $k < t$ we study the packing chromatic number of infinite distance graphs $D(k, t)$, i.e. graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in \{k, t\}$. We generalize results by Ekstein et al. for graphs $D (1, t)$. For sufficiently large $t$ we prove that $χ_ρ(D(k, t)) \leq 30$ for both $k$, $t$ odd, and that $χ_ρ(D(k, t)) \leq 56$ for exactly one of $k$, $t$ odd. We also give some upper and lower bounds for $χ_ρ(D(k, t))$ with small $k$ and $t$. Keywords: distance graph; packing coloring; packing chromatic number

math.CO↗
arXiv2012

Star subdivisions and connected even factors in the square of a graph

For any positive integer $s$, a $[2,2s]$-factor in a graph $G$ is a connected even factor with maximum degree at most $2s$. We prove that if every induced $S(K_{1, 2s+1})$ in a graph $G$ has at least 3 edges in a block of degree at most two, then $G^2$ has a $[2,2s]$-factor. This extends the results of Hendry and Vogler and of Abderrezzak et al.

math.CO↗
arXiv2012

Hamiltonian Cycles in the Square of a Graph

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.

math.CO↗
arXiv2011

The rainbow connection number of 2-connected graphs

The rainbow connection number of a graph G is the least number of colours in a (not necessarily proper) edge-colouring of G such that every two vertices are joined by a path which contains no colour twice. Improving a result of Caro et al., we prove that the rainbow connection number of every 2-connected graph with n vertices is at most the ceiling of n/2. The bound is optimal.

math.CO↗
arXiv2011

Packing Chromatic Number of Distance Graphs

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that vertices of $G$ can be partitioned into disjoint classes $X_1, ..., X_k$ where vertices in $X_i$ have pairwise distance greater than $i$. We study the packing chromatic number of infinite distance graphs $G(Z, D)$, i.e. graphs with the set $Z$ of integers as vertex set and in which two distinct vertices $i, j \in Z$ are adjacent if and only if $|i - j| \in D$. In this paper we focus on distance graphs with $D = \{1, t\}$. We improve some results of Togni who initiated the study. It is shown that $χ_ρ(G(Z, D)) \leq 35$ for sufficiently large odd $t$ and $χ_ρ(G(Z, D)) \leq 56$ for sufficiently large even $t$. We also give a lower bound 12 for $t \geq 9$ and tighten several gaps for $χ_ρ(G(Z, D))$ with small $t$.

cs.DM↗
arXiv2010

The packing chromatic number of the square lattice is at least 12

The packing chromatic number $χ_ρ(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set $V(G)$ can be partitioned into disjoint classes $X_1, ..., X_k$, where vertices in $X_i$ have pairwise distance greater than $i$. For the 2-dimensional square lattice $\mathbb{Z}^2$ it is proved that $χ_ρ(\mathbb{Z}^2) \geq 12$, which improves the previously known lower bound 10.

cs.DM↗