Related papers: Two floor building needing eight colors
We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and…
In a fractional coloring, vertices of a graph are assigned measurable subsets of the real line and adjacent vertices receive disjoint subsets; the fractional chromatic number of a graph is at most $k$ if it has a fractional coloring in…
We provide a simplified proof of the following special case of Wegner's conjecture: every planar graph of maximum degree at most three admits a distance-2 coloring with at most eight colors. Our main contribution is significant…
Two inequalities bridging the three isolated graph invariants, incidence chromatic number, star arboricity and domination number, were established. Consequently, we deduced an upper bound and a lower bound of the incidence chromatic number…
The clustered chromatic number of a graph class is the minimum integer $t$ such that for some $C$ the vertices of every graph in the class can be colored in $t$ colors so that every monochromatic component has size at most $C$. We show that…
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…
We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by…
An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part…
For an edge-colored graph $G$, the minimum color degree of $G$ means the minimum number of colors on edges which are adjacent to each vertex of $G$. We prove that if $G$ is an edge-colored graph with minimum color degree at least $5$ then…
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…
Irregular computations on unstructured data are an important class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The…
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 $\Delta_{\alpha}$…
Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In…
We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their…
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $\omega$ has chromatic number at most $3\cdot…
The mutual-visibility chromatic number of a graph $G$ is the smallest number of colors needed to color the vertices of $G$ such that each color class is a mutual-visibility set. In this paper, we prove that determining the mutual-visibility…
We consider the number of colors for the colorings of links by the symmetric group $S_3$ of degree $3$. For knots, such a coloring corresponds to a Fox 3-coloring, and thus the number of colors must be 1 or 3. However, for links, there are…
A $(a,b)$-coloring of a graph $G$ associates to each vertex a $b$-subset of a set of $a$ colors in such a way that the color-sets of adjacent vertices are disjoint. We define general reduction tools for $(a,b)$-coloring of graphs for $2\le…
The Four Colour Theorem asserts that the vertices of every plane graph can be properly coloured with four colors. Fabrici and G\"oring conjectured the following stronger statement to also hold: the vertices of every plane graph can be…
A triangulation of a polygon is a subdivision of it into triangles, using diagonals between its vertices. Two different triangulations of a polygon can be related by a sequence of flips: a flip replaces a diagonal by the unique other…