Improved square coloring of planar graphs

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac{3}{2}\mathrm{\Delta }+1$ colors are sufficient to square color every planar graph of maximum degree Δ. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2\mathrm{\Delta }+7$ colors are sufficient, which improves the best known bounds when $6?\mathrm{\Delta }?31$.     

Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

The impact of loops on the game of cops and robbers on graphs

In the game of cops and robbers on graphs, the cops and the robber are allowed to pass their turn if they are located on a looped vertex. This paper explores the effect of loops on the cop number and the capture time. We provide examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play.     

Factorization method and new potentials from the inverted oscillator

In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.     

The eigenpairs of a Sylvester–Kac type matrix associated with a simple model for one-dimensional deposition and evaporation

A straightforward model for deposition and evaporation on discrete cells of a finite array of any dimension leads to a matrix equation involving a Sylvester–Kac type matrix. The eigenvalues and eigenvectors of the general matrix are determined for an arbitrary number of cells. A variety of models to which this solution may be applied are discussed.     

Vertex partitions of (C3,C4,C6)-free planar graphs

A graph is $(k_1,k_2)$-colorable if it admits a vertex partition into a graph with maximum degree at most $k_1$ and a graph with maximum degree at most $k_2$. We show that every $(C_3,C_4,C_6)$-free planar graph is $(0,6)$-colorable. We also show that deciding whether a $(C_3,C_4,C_6)$-free planar graph is $(0,3)$-colorable is NP-complete.     

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension $n?1$ in ${\mathbb{R}}^n$, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in ${\mathbb{R}}^n$, $d<n?1$, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ${\omega }_L$ is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ${\omega }_L$ and the Hausdorff measure.