Related papers: On Chromatic Core Subgraph of Simple Graphs

The main goal of this paper is to formalize and explore a connection between chromatic properties of graphs with geometric representations and competitive analysis of on-line algorithms, which became apparent after the recent construction…

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to color) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color…

We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of…

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most one. The equitable chromatic number of a graph $G$, denoted by $\chi_=(G)$, is the minimum $k$ such that $G$ is equitably $k$-colorable. The…

For a hypergraph $G$, let $\chi(G), \Delta(G),$ and $\lambda(G)$ denote the chromatic number, the maximum degree, and the maximum local edge connectivity of $G$, respectively. A result of Rhys Price Jones from 1975 says that every connected…

We consider the $t$-improper chromatic number of the Erd{\H o}s-R{\'e}nyi random graph $G(n,p)$. The t-improper chromatic number $\chi^t(G)$ of $G$ is the smallest number of colours needed in a colouring of the vertices in which each colour…

A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other…

Given an arbitrary graph $G$ we study the chromatic number of a random subgraph $G_{1/2}$ obtained from $G$ by removing each edge independently with probability $1/2$. Studying $\chi(G_{1/2})$ has been suggested by Bukh~\cite{Bukh}, who…

Given a graph or multigraph $G$, let $\chi'_{trans}(G)$ denote the minimum integer $n$ such that any proper $\chi'(G)$--edge coloring of $G$ can be transformed into any other proper $\chi'(G)$--edge coloring of $G$ by a series of…

The chromatic edge stability index $\mathrm{es}_{\chi'}(G)$ of a graph $G$ is the minimum number of edges whose removal results in a graph with smaller chromatic index. We give best-possible upper bounds on $\mathrm{es}_{\chi'}(G)$ in terms…

A proper vertex colouring of a graph $G$ is referred to as conflict-free if in the neighbourhood of every vertex some colour appears exactly once, while it is called $h$-conflict-free if there are at least $h$ such colours for each vertex…

The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes -- a purely combinatorial one and two geometric ones. It is shown, that most…

A coloring of a connected graph $G$ is a function $f$ mapping the vertex set of $G$ into the set of all integers. For any subgraph $H$ of $G$, we denote the sum of the values of $f$ on the vertices of $H$ as $f(H)$. If for any integer $k\in…

Let $\chi'_\subset(G)$ be the least number of colours necessary to properly colour the edges of a graph $G$ with minimum degree $\delta\geq 2$ so that the set of colours incident with any vertex is not contained in a set of colours incident…

For a fixed graph $H$, in the List $H$-Coloring problem, we are given a graph $G$ along with list $L(v) \subseteq V(H)$ for every $v \in V(G)$, and we have to determine if there exists a list homomorphism $\varphi$ from $(G,L)$ to $H$,…

In the online graph coloring problem, vertices from a graph G, known in advance, arrive in an online fashion and an algorithm must immediately assign a color to each incoming vertex v so that the revealed graph is properly colored. The…

Resolving a problem raised by Norin, we show that for each $k \in \mathbb{N}$, there exists an $f(k) \le 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic…

Let $G=(V(G), E(G))$ be a graph with maximum degree $\Delta$. For a subset $M$ of $E(G)$, we denote by $G[V(M)]$ the subgraph of $G$ induced by the endvertices of edges in $M$. We call $M$ a semistrong matching if each edge of $M$ is…

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $\chi_{\rm pcf}^h(G)$ denote the minimum $k$ such that…

With respect to a proper colouring of a graph $G$, we know that $\delta(G) \leq \chi(G) \leq \Delta(G)+1$. If distinct colours represent distinct technology types to be located at vertices the question arises on how to place at least one of…