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One of the questions in Rigidity Theory is whether a realization of the vertices of a graph in the plane is flexible, namely, if it allows a continuous deformation preserving the edge lengths. A flexible realization of a connected graph in…
For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings.…
We consider realizations of a graph in the plane such that the distances between adjacent vertices satisfy the constraints given by an edge labeling. If there are infinitely many such realizations, counted modulo rigid motions, the labeling…
We study the existence of an $n$-fold rotationally symmetric placement of a symmetric graph in the plane allowing a continuous deformation that preserves the symmetry and the distances between adjacent vertices. We show that such a flexible…
Proper graph coloring assigns different colors to adjacent vertices of the graph. Usually, the number of colors is fixed or as small as possible. Consider applications (e.g. variants of scheduling) where colors represent limited resources…
We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our…
A walk $W$ in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in $W$ is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity…
An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…
Given a periodic graph, we wish to determine via combinatorial methods whether it has periodic embeddings in the plane that -- via motions that preserve edge-lengths and periodicity -- can be continuously deformed into another non-congruent…
A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a cyclic interval $t$-coloring if for each vertex $v$ of $G$ the edges incident to $v$ are colored by consecutive colors, under the condition that color $1$ is…
It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For…
A proper edge-coloring of a graph $G$ with colors $1,\ldots,t$ is called an \emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo…
An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and…
An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycle s. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic e dge coloring using k colors…
A graph is (7, 2)-edge-choosable if, for every assignment of lists of size 7 to the edges, it is possible to choose two colors for each edge from its list so that no color is chosen for two incident edges. We show that every…
A (2-dimensional) realisation of a graph $G$ is a pair $(G,p)$, where $p$ maps the vertices of $G$ to $\mathbb{R}^2$. A realisation is flexible if it can be continuously deformed while keeping the edge lengths fixed, and rigid otherwise. We…
NAC-colourings of graphs correspond to flexible quasi-injective realisations in $\mathbb {R} ^2$. A special class of NAC-colourings are those that arise from stable cuts. We give sharp thresholds for the random graph to have no stable cut…
An edge colouring of a graph is called distinguishing if there is no non-trivial automorphism which preserves it. We prove that every at most countable, finite or infinite, connected regular graph of order at least $7$ admits a…
A proper edge coloring of a graph $G$ is called acyclic if there is no bichromatic cycle in $G$. The acyclic chromatic index of $G$, denoted by $\chi'_a(G)$, is the least number of colors $k$ such that $G$ has an acyclic edge $k$-coloring.…
\textit{A star edge coloring} of a graph is a proper edge coloring without bichromatic paths and cycles of length four. In this paper we establish tight upper bounds for trees and subcubic outerplanar graphs, and derive an upper bound for…