Related papers: Note on polychromatic coloring of hereditary hyper…

We popularize the question whether, for $m$ large enough, all $m$-uniform shift-chain hypergraphs are properly $2$-colorable. On the other hand, we show that for every $m$ some $m$-uniform shift-chains are not polychromatic $3$-colorable.

Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$…

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many…

For a number $\ell\geq 2$, let $\mathcal{H}_{\ell}$ denote the family of graphs which have girth $2\ell$ and have no even hole with length greater than $2\ell$. Wu, Xu, and Xu conjectured that every graph in…

A coloured graph is k-ultrahomogeneous if every isomorphism between two induced subgraphs of order at most k extends to an automorphism. A coloured graph is t-tuple regular if the number of vertices adjacent to every vertex in a set S of…

Let $G=(V,E)$ be a simple graph and $(2k+1)$ be a prime integer. Let each vertex of $G$ be colored using one of the $(2k+1)$ colors, say $R_1,R_2,...,R_{2k+1}$. If every vertex has an equal number of neighbors of each color, then the…

For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as…

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs…

For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much…

A hypergraph $\mathcal{H}$ on $n$ vertices and $m$ edges is said to be {\it nearly-intersecting} if every edge of $\mathcal{H}$ intersects all but at most polylogarthmically many (in $m$ and $n$) other edges. Given lists of colors…

A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2\lfloor \sqrt{k}\rfloor)$-colorable $k$-uniform hypergraph, it is…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

For every positive integer $n$, we construct a Hasse diagram with $n$ vertices and chromatic number $\Omega(n^{1/4})$, which significantly improves on the previously known best constructions of Hasse diagrams having chromatic number…

In this paper, we study the homology of the coloring complex and the cyclic coloring complex of a complete $k$-uniform hypergraph. We show that the coloring complex of a complete $k$-uniform hypergraph is shellable, and we determine the…

Let $H$ be a fixed graph. Denote $f(n,H)$ to be the maximum number of edges not contained in any monochromatic copy of $H$ in a 2-edge-coloring of the complete graph $K_n$, and $ex(n,H)$ to be the {\it Tur\'an number} of $H$. An easy lower…

Proper conflict-free coloring is an intermediate notion between proper coloring of a graph and proper coloring of its square. It is a proper coloring such that for every non-isolated vertex, there exists a color appearing exactly once in…

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general…

We study hypergraphs which are uniquely determined by their chromatic, independence and matching polynomials. B. Bollob\'as, L. Pebody and O. Riordan (2000) conjectured (BPR-conjecture) that almost all graphs are uniquely determined by…

Given a hypergraph H = (V, E), a coloring of its vertices is said to be conflict-free if for every hyperedge S \in E there is at least one vertex in S whose color is distinct from the colors of all other vertices in S. The discrete interval…

Let G = (V, E) be a multigraph without loops and for any x {\in}V let E(x) be the set of edges of G incident to x. A homogeneous edge-coloring of G is an assignment of an integer m >= 2 and a coloring c:E {\to} S of the edges of…