Related papers: $2$-distance $(\Delta+1)$-coloring of sparse graph…
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 proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$.…
In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$: For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this is a proper colouring of the vertices of $G$ such that, in addition, vertices that…
A {\em conflict-free coloring} of a graph {\em with respect to open} (resp., {\em closed}) {\em neighborhood} is a coloring of vertices such that for every vertex there is a color appearing exactly once in its open (resp., closed)…
A vertex coloring of a graph $G$ is an assignment of colors to the vertices of $G$ such that every two adjacent vertices of $G$ have different colors. A coloring related property of a graphs is also an assignment of colors or labels to the…
Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $\Delta+1$ colors, where $\Delta$ denotes the maximum degree. Using $\Delta+1$ colors may be…
Erd\H{o}s, Pach, Pollack and Tuza [J. Combin. Theory, B 47, (1989), 279-285] conjectured that the diameter of a $K_{2r}$-free connected graph of order $n$ and minimum degree $\delta\geq 2$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)}\cdot…
The "separation dimension" of a graph $G$ is the minimum positive integer $d$ for which there is an embedding of $G$ into $\mathbb{R}^d$, such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a…
A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} $\chiup_{s}'(G)$ of a graph $G$ is the minimum number of colors in a strong edge…
A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P.…
We study vertex colourings of digraphs so that no out-neighbourhood is monochromatic and call such a colouring an {\bf out-colouring}. The problem of deciding whether a given digraph has an out-colouring with only two colours (called a…
The acyclic chromatic number of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. We show that for all $\alpha>2^{-1/3}$ there exists an integer $\Delta_{\alpha}$…
The Chen-Lih-Wu Conjecture states that each connected graph with maximum degree $\Delta\geq 3$ that is not the complete graph $K_{\Delta+1}$ or the complete bipartite graph $K_{\Delta,\Delta}$ admits an equitable coloring with $\Delta$…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We…
The famous Wegner's Planar Graph Conjecture asserts tight upper bounds on the chromatic number of the square $G^2$ of a planar graph $G$, depending on the maximum degree $\Delta(G)$ of $G$. The only case that the conjecture is resolved is…
Given a perfect coloring of a graph, we prove that the $L_1$ distance between two rows of the adjacency matrix of the graph is not less than the $L_1$ distance between the corresponding rows of the parameter matrix of the coloring. With the…
A $t$-tone coloring of a graph $G$ assigns to each vertex a set of $t$ colors such that any pair of vertices $u, v$ with distance $d$ can share at most $d-1$ colors. In this note, we prove several new results on $t$-tone coloring. For…
We propose the notion of a majority $k$-edge-coloring of a graph $G$, which is an edge-coloring of $G$ with $k$ colors such that, for every vertex $u$ of $G$, at most half the edges of $G$ incident with $u$ have the same color. We show the…
The $2$-layer drawing model is a well-established paradigm to visualize bipartite graphs. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of $k$-planar graphs has been…
This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof…