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The chromatic number for properly colouring the facets of a combinatorial simple $n$-polytope $P^n$ that is the orbit space of a quasitoric manifold satisfies the inequality $n\leq P^n\leq 2^n-1$. The inequality is sharp for $n=2$ but not…
We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset…
In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G…
A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
Let $G$ be a graph such that each edge has its list of available colors, and assume that each list is a subset of the common set consisting of $k$ colors. Suppose that we are given two list edge-colorings $f_0$ and $f_r$ of $G$, and asked…
Given a connected graph $G$ and its vertex $x$, let $U_x(G)$ denote the universal cover of $G$ obtained by unfolding $G$ into a tree starting from $x$. Let $T=T(n)$ be the minimum number such that, for graphs $G$ and $H$ with at most $n$…
In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…
A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…
We consider semilinear elliptic problems on two-dimensional hyperbolic space involving critical growth. We first establish the Palais-Smale(P-S) condition and using (P-S) condition we obtain existence of solutions. In addition, we also…
The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly,…
We deal with the general problem of scattering by open-arcs in two-dimensional space. We show that this problem can be solved by means of certain second-kind integral equations of the form $\tilde{N} \tilde{S}[\varphi] = f$, where…
We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the projective plane, with the single exception of the Petersen graph, is 3-edge-colorable. In other words, the only (non-trivial) snark that can be…
We propose a new exact approach to the generalized graph layering problem that is based on a particular quadratic assignment formulation. It expresses, in a natural way, the associated layout restrictions and several possible objectives,…
We present a simple approach for purifying graphite surface and vacancy states using an angle-resolved x-ray photoelectron residual spectroscopy (XPS). Complementing the discoveries of Ugeda et al [Phys Rev Lett 104, 096804 (2010)],…
A $k$-bisection of a bridgeless cubic graph $G$ is a $2$-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic…
In this paper we study colorings (or tilings) of the two-dimensional grid $\mathbb{Z}^2$. A coloring is said to be valid with respect to a set $P$ of $n\times m$ rectangular patterns if all $n\times m$ sub-patterns of the coloring are in…
This paper presents two approaches to reducing problems on $2$-cycles on a smooth cubic hypersurface $X$ over an algebraically closed field of characteristic $\neq 2$, to problems on $1$-cycles on its variety of lines $F(X)$. The first one…
A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. If the coloring is required to be proper, then the upper bound for the…
We present a theoretical method to study driven-dissipative correlated systems on lattices with two spatial dimensions (2D). The steady-state density-matrix of the lattice is obtained by solving the master equation in a corner of the…