Related papers: Edge Cover Colouring Versus Minimum Degree in Mult…
A strong edge-colouring of a graph is a proper edge-colouring where each colour class induces a matching. It is known that every planar graph with maximum degree $\Delta$ has a strong edge-colouring with at most $4\Delta+4$ colours. We show…
We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the…
An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from…
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose edges have distinct colors. The \textit{color degree} of a vertex $v$ is the number of different colors on edges incident to $v$. We show that if $n$ is large enough…
Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum number of colors so that no two adjacent edges have the same color. This problem can be naturally formulated as an integer…
A matching $M$ in a graph $G$ is {\em semistrong} if every edge of $M$ has an endvertex of degree one in the subgraph induced by the vertices of $M$. A {\em semistrong edge-coloring} of a graph $G$ is a proper edge-coloring in which every…
In 1985, Erd\H{o}s and Ne\'{s}etril conjectured that the strong edge-coloring number of a graph is bounded above by ${5/4}\Delta^2$ when $\Delta$ is even and ${1/4}(5\Delta^2-2\Delta+1)$ when $\Delta$ is odd. They gave a simple construction…
The classic theorem of Vizing (Diskret. Analiz.'64) asserts that any graph of maximum degree $\Delta$ can be edge colored (offline) using no more than $\Delta+1$ colors (with $\Delta$ being a trivial lower bound). In the online setting,…
A strong edge-coloring of a graph $G$ is a coloring of the edges such that every color class induces a matching in $G$. The strong chromatic index of a graph is the minimum number of colors needed in a strong edge-coloring of the graph. In…
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced…
An \emph{interval $t$-coloring} of a multigraph $G$ is a proper edge coloring with colors $1,\dots,t$ such that the colors on the edges incident to every vertex of $G$ are colored by consecutive colors. A \emph{cyclic interval $t$-coloring}…
Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing…
We prove asymptotically optimal bounds on the number of edges a graph $G$ must have in order that any $r$-colouring of $E(G)$ has a colour class which contains every $D$-degenerate graph on $n$ vertices with bounded maximum degree. We also…
In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an $n$-vertex edge-colored graph $G$ with colors from $\{1,…
A graph $G$ is class II, if its chromatic index is at least $\Delta+1$. Let $H$ be a maximum $\Delta$-edge-colorable subgraph of $G$. The paper proves best possible lower bounds for $\frac{|E(H)|}{|E(G)|}$, and structural properties of…
An edge-face colouring of a plane graph with edge set $E$ and face set $F$ is a colouring of the elements of $E \cup F$ such that adjacent or incident elements receive different colours. Borodin proved that every plane graph of maximum…
A {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same…
A vertex colouring of a graph is called asymmetric if the only automorphism which preserves it is the identity. Tucker conjectured that if every automorphism of a connected, locally finite graph moves infinitely many vertices, then there is…
An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…
An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an integer interval. It is well-known that…