Related papers: Signed projective cubes, a homomorphism point of v…
We study a new variant of \emph{connected coloring} of graphs based on the concept of \emph{strong} edge coloring (every color class forms an \emph{induced} matching). In particular, an edge-colored path is \emph{strongly proper} if its…
A cube-like graph is a Cayley graph for the elementary abelian group of order $2^n$. In studies of the chromatic number of cube-like graphs, the $k$th power of the $n$-dimensional hypercube, $Q_n^k$, is frequently considered. This coloring…
Taking the Levine-Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau…
In an edge-coloring of a cubic graph, an edge is poor or rich, if the set of colors assigned to the edge and the four edges adjacent it, has exactly five or exactly three distinct colors, respectively. An edge is normal in an edge-coloring…
We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges…
Mirror graphs were introduced by Bre\v{s}ar et al. in 2004 as an intriguing class of graphs: vertex-transitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article…
A normal 5-edge-coloring of a cubic graph is a coloring such that for every edge the number of distinct colors incident to its end-vertices is 3 or 5 (and not 4). The well known Petersen Coloring Conjecture is equivalent to the statement…
A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique…
We extend the theory of circular game chromatic numbers to signed graphs by defining the invariant $\chi_c^g(G,\sigma)$ for signed graphs $(G,\sigma)$. Our analysis establishes tight bounds dependent on the structural properties of the…
We study the number of (set-theoretically) defining equations of Segre products of projective spaces times certain projective hypersurfaces, extending results by Singh and Walther. Meanwhile, we prove some results about the cohomological…
The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…
In a proper edge-coloring of a cubic graph, an edge $e$ is normal if the set of colors used by the edges adjacent to $e$ has cardinality 3 or 5. The Petersen coloring conjecture asserts that every bridgeless cubic graph has a normal…
We extend the notion of graph homomorphism to cellularly embedded graphs (maps) by designing operations on vertices and edges that respect the surface topology; we thus obtain the first definition of map homomorphism that preserves both the…
We prove that every cyclically 4-edge-connected cubic graph that can be embedded in the torus, with the exceptional graph class called "Petersen-like", is 3-edge-colorable. This means every (non-trivial) toroidal snark can be obtained from…
A middle-cube is an induced subgraph consisting of nodes at the middle two layers of a hypercube. The middle-cubes are related to the well-known Revolving Door (Middle Levels) conjecture. We study the middle-cube graph by completely…
A class of surfaces-graphs in a Riemannian 3-space with a prescribed projection of one field of principal directions onto a surface $\Pi$ is considered. A problem of determination of such surfaces when both principal curvatures are given…
The pseudo-Grundy index of a graph is the largest number of colors that can be assigned to its edges, such that for every pair of colors $i,j$, if $i < j$ then every edge colored with color $j$ is adjacent to at least one edge colored with…
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if…
We investigate properties of signed graphs that have few distinct eigenvalues together with a symmetric spectrum. Our main contribution is to determine all signed $(0,2)$-graphs with vertex degree at most $6$ that have precisely two…
In 1982, Zaslavsky introduced the concept of a proper vertex colouring of a signed graph $G$ as a mapping $\phi\colon V(G)\to \mathbb{Z}$ such that for any two adjacent vertices $u$ and $v$ the colour $\phi(u)$ is different from the colour…