On tours that contain all edges of a hypergraph.
On traceability and 2-factors in claw-free graphs
If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σₖ > n + k² - 4k + 7 (where k is an arbitrary constant), then G has a 2-factor with at most k - 1 components. As a second main result, we present classes of graphs ₁,...,₈ such that every sufficiently large connected claw-free graph satisfying degree condition σ₆(k) > n + 19 (or, as a corollary, δ(G) > (n+19)/6) either belongs to or is traceable.
On transitive orientations of G-ê
A comparability graph is a graph whose edges can be oriented transitively. Given a comparability graph G = (V,E) and an arbitrary edge ê∈ E we explore the question whether the graph G-ê, obtained by removing the undirected edge ê, is a comparability graph as well. We define a new substructure of implication classes and present a complete mathematical characterization of all those edges.
On Translating One Polyomino To Tile the Plane.
On translation transversal designs
On tree-complete graphs
On Twin Edge Colorings of Graphs
A twin edge k-coloring of a graph G is a proper edge coloring of G with the elements of Zk so that the induced vertex coloring in which the color of a vertex v in G is the sum (in Zk) of the colors of the edges incident with v is a proper vertex coloring. The minimum k for which G has a twin edge k-coloring is called the twin chromatic index of G. Among the results presented are formulas for the twin chromatic index of each complete graph and each complete bipartite graph
On two- and three-element subsets of groups.
On two combinatorial identities
On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices.
On two graph-theoretical problems from the conference at Nová Ves u Branžeže
On two problems of the graph theory
On two problems regarding the Hamiltonian cycle game.
On two supercongruences involving Almkvist-Zudilin sequences
We prove two supercongruences involving Almkvist-Zudilin sequences, which were originally conjectured by Z.-H. Sun (2020).
On two transformations of graphs
The paper describes the properties of two transformations of graphs. One of them was introduced by F. Gliviak for the sake of study of metric properties of graphs, the other is related to it.
On Ulam's hypothesis
On unimodality problems in Pascal's triangle.
On Unique Minimum Dominating Sets in Some Cartesian Product Graphs
Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set.
On uniquely colorable graphs without short cycles
On Uniquely Hamiltonian Claw-Free and Triangle-Free Graphs
A graph is uniquely Hamiltonian if it contains exactly one Hamiltonian cycle. In this note, we prove that claw-free graphs with minimum degree at least 3 are not uniquely Hamiltonian. We also show that this is best possible by exhibiting uniquely Hamiltonian claw-free graphs with minimum degree 2 and arbitrary maximum degree. Finally, we show that a construction due to Entringer and Swart can be modified to construct triangle-free uniquely Hamiltonian graphs with minimum degree 3.