Related papers: A quantum protocol to win the graph colouring game…
The study of quantum chromatic numbers of graphs is a hot research topic in recent years. However, the infinite family of graphs with known quantum chromatic numbers are rare, as far as we know, the only known such graphs (except for…
The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in…
Given a graph $G$ and $k \in \mathbb{N}$, we introduce the following game played in $G$. Each round, Alice colours an uncoloured vertex of $G$ red, and then Bob colours one blue (if any remain). Once every vertex is coloured, Alice wins if…
The Colouring problem asks whether the vertices of a graph can be coloured with at most $k$ colours for a given integer $k$ in such a way that no two adjacent vertices receive the same colour. A graph is $(H_1,H_2)$-free if it has no…
We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical…
We define a family of vertex colouring games played over a pair of graphs or digraphs $(G,H)$ by players $\forall$ and $\exists$. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is…
We combine the ideas of edge coloring games and asymmetric graph coloring games and define the \emph{$(m,1)$-edge coloring game}, which is alternatively played by two players Maker and Breaker on a finite simple graph $G$ with a set of…
For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…
We present an architecture for early fault-tolerant quantum computers based on the smallest interesting colour code (Earl Campbell, 2016). It realizes a universal logical gate set consisting of single-qubit measurements and preparations in…
A weak odd dominated (WOD) set in a graph is a subset B of vertices for which there exists a distinct set of vertices C such that every vertex in B has an odd number of neighbors in C. We point out the connections of weak odd domination…
The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to vertices of a graph such that no two adjacent vertices receive the same color. GCP has been…
We study the two-player game where Maker and Breaker alternately color the edges of a given graph $G$ with $k$ colors such that adjacent edges never get the same color. Maker's goal is to play such that at the end of the game, all edges are…
In this note, we present results for the colouring problem on small world graphs created by rewiring square, triangular, and two kinds of cubic (with coordination numbers 5 and 6) lattices. As the rewiring parameter p tends to 1, we find…
Assume $n$ players are placed on the $n$ vertices of a graph $G$. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a colour out of $q$ available colours. The players can see the hat of…
A $q$-\emph{equitable coloring} of a graph $G$ is a proper $q$-coloring such that the sizes of any two color classes differ by at most one. In contrast with ordinary coloring, a graph may have an equitable $q$-coloring but has no equitable…
As early as 1935, Schr\"odinger recognized entanglement as ``not one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought''. Indeed, most remarkable phenomena…
An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by…
A $K_3$-WORM coloring of a graph $G$ is an assignment of colors to the vertices in such a way that the vertices of each $K_3$-subgraph of $G$ get precisely two colors. We study graphs $G$ which admit at least one such coloring. We disprove…
The hat guessing number $HG(G)$ of a graph $G$ on $n$ vertices is defined in terms of the following game: $n$ players are placed on the $n$ vertices of $G$, each wearing a hat whose color is arbitrarily chosen from a set of $q$ possible…
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…