Related papers: Minimal obstructions to $C_5$-coloring in heredita…
The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color the vertices of $G$ such that the only color preserving automorphism is the identity. For infinite graphs $D(G)$ is bounded by the…
A graph coloring has bounded clustering if each monochromatic component has bounded size. Equivalently, it is a partition of the vertices into induced subgraphs with bounded size components. This paper studies clustered colorings of graphs,…
An edge coloring of a graph $G$ is called conflict-free if, for every edge, its closed neighborhood contains a color that appears exactly once. The least number of colors required for such a coloring is the conflict-free chromatic index of…
In this paper, we prove that every 3-chromatic connected graph, except $C_7$, admits a 3-vertex coloring in which every vertex is the beginning of a 3-chromatic path. It is a special case of a conjecture due to S.~Akbari, F.~Khaghanpoor,…
A graph is $(c_1, c_2, ..., c_k)$-colorable if the vertex set can be partitioned into $k$ sets $V_1,V_2, ..., V_k$, such that for every $i: 1\leq i\leq k$ the subgraph $G[V_i]$ has maximum degree at most $c_i$. We show that every planar…
Inspired by the majority colorings and C-colorings, we introduce and study the majority C-coloring of graphs. In such a vertex coloring, every vertex shares its color with at least half of its neighbors. The maximum number of colors that…
We relate star colouring of even-degree regular graphs to the notions of locally constrained graph homomorphisms to the oriented line graph $ \vec{L}(K_q) $ of the complete graph $ K_q $ and to its underlying undirected graph $ L^*(K_q) $.…
We look at colourings of $r$-uniform hypergraphs, focusing our attention on unique colourability and gaps in the chromatic spectrum. The pattern of an edge $E$ in an $r$-uniform hypergraph $H$ whose vertices are coloured is the partition of…
A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…
An undirected graph G is called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at most h is denoted by [h,2,1]. The…
We describe a simple homological test for obstructions to graph colorings. The main idea is to combine the framework of Hom-complexes with the following general fact: an arbitrary Z_2-space has nontrivial homology with Z_2-coefficients in…
We consider cell colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a cell $k$-coloring of $\Gamma(G)$ to be a coloring of the…
A complete $k$-coloring of a graph $G=(V,E)$ is an assignment $\varphi:V\to\{1,\ldots,k\}$ of colors to the vertices such that no two vertices of the same color are adjacent, and the union of any two color classes contains at least one…
The topological containment problem is known to be polynomial-time solvable for any fixed pattern graph $H$, but good characterisations have been found for only a handful of non-trivial pattern graphs. The complete graph on five vertices,…
A proper edge coloring of a graph $G$ with colors $1,2,\dots,t$ is called a \emph{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…
Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order $\Omega(\frac{n}{\log(n)})$.…
The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered…
We say a class $\mathcal{C}$ of graphs is clean if for every positive integer $t$ there exists a positive integer $w(t)$ such that every graph in $\mathcal{C}$ with treewidth more than $w(t)$ contains an induced subgraph isomorphic to one…
The class of even-hole-free graphs has been extensively studied on its own and on its relation to perfect graphs. In this paper, we study the $\chi$-boundedness of even-hole-free graphs which itself is an important topic in graph theory. In…
Given a graph $G=(V,E)$ whose vertices have been properly coloured, we say that a path in $G$ is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly…