Related papers: Total Colourings of Direct Product Graphs
A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various…
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…
A proper vertex-coloring of a graph is $r$-dynamic if the neighbors of each vertex $v$ receive at least $\min(r, \mathrm{deg}(v))$ different colors. In this note, we prove that if $G$ has a strong $2$-coloring number at most $k$, then $G$…
For a fixed number of colors, we show that, in node-weighted split graphs, cographs, and graphs of bounded tree-width, one can determine in polynomial time whether a proper list-coloring of the vertices of a graph such that the total weight…
A proper vertex coloring of a graph is a mapping of its vertices on a set of colors, such that two adjacent vertices are not mapped to the same color. This constraint may be interpreted in terms of the distance between to vertices and so a…
In this paper, we consider a weighted generalization of the chromatic number of a Binomial random graph~\(G.\) We equip each edge with a random weight and then colour the vertices in such a way that the absolute colour difference between…
A k-fold x-coloring of a graph is an assignment of (at least) k distinct colors from the set {1, 2, ..., x} to each vertex such that any two adjacent vertices are assigned disjoint sets of colors. The smallest number x such that G admits a…
In this paper, we present a foundation study for proper colouring of edge-set graphs. The authors consider that a detailed study of the colouring of edge-set graphs corresponding to the family of paths is best suitable for such foundation…
Total colouring of 4-regular circulant graphs is an interesting but challenging topic, and has attracted much attention. However, it still remains an open question to determine the total chromatic numbers of $C_{n}(1, 3)$, a subclass of…
If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest…
We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…
A majority coloring of an undirected graph is a vertex coloring in which for each vertex there are at least as many bi-chromatic edges containing that vertex as monochromatic ones. It is known that for every countable graph a majority…
In this work we show that, for any fixed d, random d-regular graphs asymptotically almost surely can be coloured with k colours, where k is the smallest integer satisfying d<2(k-1)log(k-1). From previous lower bounds due to Molloy and Reed,…
For a simple graph G = (V, E) and a positive integer k greater than or equal to 2, a coloring of vertices of G using exactly k colors such that every vertex has an equal number of vertices of each color in its closed neighborhood is called…
A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a…
A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called "beyond planar graphs". A…
A graph is 1-planar if it can be drawn on a plane so that each edge is crossed by at most one other edge. In this paper, we first give a useful structural theorem for 1-planar graphs, and then apply it to the list edge and list total…
We introduce the total dominator edge chromatic number of a graph $G$. A total dominator edge coloring (briefly TDE-coloring) of $G$ is a proper edge coloring of $G$ in which each edge of the graph is adjacent to every edge of some color…
We confirm the equitable $\Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or…
A graph is called $k$-extendable if each $k$-matching can be extended to a perfect matching. We give spectral conditions for the $k$-extendability of graphs and bipartite graphs using Tutte-type and Hall-type structural characterizations.…