Related papers: Colouring powers of cycles from random lists

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollob\'as conjectured that for every \epsilon>0 and positive…

A clique colouring of a graph is a colouring of the vertices such that no maximal clique is monochromatic (ignoring isolated vertices). The least number of colours in such a colouring is the clique chromatic number. Given $n$ points $x_1,…

Let $K_n^{(k)}$ be the complete $k$-graph on $n$ vertices. A $k$-uniform tight cycle is a $k$-graph with its vertices cyclically ordered so that every $k$ consecutive vertices form an edge and any two consecutive edges share exactly $k-1$…

We deal with $k$-out-regular directed multigraphs with loops (called simply \emph{digraphs}). The edges of such a digraph can be colored by elements of some fixed $k$-element set in such a way that outgoing edges of every vertex have…

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…

We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices.…

In this paper, the on-line list colouring of binomial random graphs G(n,p) is studied. We show that the on-line choice number of G(n,p) is asymptotically almost surely asymptotic to the chromatic number of G(n,p), provided that the average…

In this note we consider a Ramsey property of random $d$-regular graphs, $\mathcal{G}(n,d)$. Let $r\ge 2$ be fixed. Then w.h.p. the edges of $\mathcal{G}(n, 2r)$ can be colored such that every monochromatic component has size $o(n)$. On the…

We study several enumeration problems connected to linear trees, a broad class which includes stars, paths, generalized stars, and caterpillars. We provide generating functions for counting the number of linear trees on $n$ vertices,…

An edge-coloured cycle is $rainbow$ if all edges of the cycle have distinct colours. For $k\geq 1$, let $\mathcal{F}_{k}$ denote the family of all graphs with the property that any $k$ vertices lie on a cycle. For $G\in \mathcal{F}_{k}$, a…

An edge colouring of a graph $G$ is called acyclic if it is proper and every cycle contains at least three colours. We show that for every $\varepsilon>0$, there exists a $g=g(\varepsilon)$ such that if $G$ has girth at least $g$ then $G$…

It is conjectured that every edge-colored complete graph $G$ on $n$ vertices satisfying $\Delta^{mon}(G)\leq n-3k+1$ contains $k$ vertex-disjoint properly edge-colored cycles. We confirm this conjecture for $k=2$, prove several additional…

Graph coloring is a computationally difficult problem, and currently the best known classical algorithm for $k$-coloring of graphs on $n$ vertices has runtimes $\Omega(2^n)$ for $k\ge 5$. The list coloring problem asks the following more…

In responding to a question on Math Stackexchange, the author formulated the problem of determining the number of strings of balls colored in most $n$ colors with a number $k$ of repeated colors. In this paper, we formulate the problem more…

We consider the k-strong conflict-free coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring has to be conflict-free, in the sense that in every…

Let ${\mathcal C}$ be a proper minor-closed family of graphs. We present a randomized algorithm that given a graph $G \in {\mathcal C}$ with $n$ vertices, finds a simple cycle of size $k$ in $G$ (if exists) in $2^{O(k)}n$ time. The…

In this paper we study different restrictions imposed over the set of permutations of size $n$, $S_n$, and for specific classes of restrictions study the cycle structure of corresponding permutations. More specifically, we prove that for…

Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you…

We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a…

Given a graph $G=(V, E)$ and a list of available colors $L(v)$ for each vertex $v\in V$, where $L(v) \subseteq \{1, 2, \ldots, k\}$, List $k$-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex…