Related papers: Schur's colouring theorem for non-commuting pairs

Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after…

If $S$ is a subset of an abelian group $G$, the polychromatic number of $S$ in $G$ is the largest integer $k$ so that there is a $k-$coloring of the elements of $G$ such that every translate of $S$ in $G$ gets all $k$ colors. We determine…

Let $G$ be a finite group, let $\pi$ be a set of primes and let $p$ be the smallest prime in $\pi$. In this work, we prove that $G$ possesses a normal and abelian Hall $\pi$-subgroup if and only if the probability that two random…

Feder and Subi conjectured that for any $2$-coloring of the edges of the $n$-dimensional cube, we can find an antipodal pair of vertices connected by a path that changes color at most once. We discuss the case of random colorings, and we…

Let $G$ be a non-abelian group. The non-commuting graph $\mathcal{A}_G$ of $G$ is defined as the graph whose vertex set is the non-central elements of $G$ and two vertices are joint if and only if they do not commute. In a finite simple…

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed…

For a digraph $G$, let $f(G)$ be the maximum chromatic number of an acyclic subgraph of $G$. For an $n$-vertex digraph $G$ it is proved that $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ where $s$ is the bipartite independence number of $G$, i.e., the…

Let $G=(V(G), E(G))$ be a multigraph with maximum degree $\Delta(G)$, chromatic index $\chi'(G)$ and total chromatic number $\chi''(G)$. The Total Coloring conjecture proposed by Behzad and Vizing, independently, states that $\chi''(G)\leq…

Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well…

The total chromatic number, $\chi''(G)$ is the minimum number of colors which need to be assigned to obtain a total coloring of the graph $G$. The Total Coloring Conjecture (TCC) made independently by Behzad and Vizing that for any graph,…

Let $G$ be a finite $2$-generated non-cyclic group. The spread of $G$ is the largest integer $k$ such that for any nontrivial elements $x_1, \ldots, x_k$, there exists $y \in G$ such that $G = \langle x_i, y\rangle$ for all $i$. The more…

Let $f: V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ be a non-proper total $k$-coloring of $G$. Define a weight function on total coloring as $$\phi(x)=f(x)+\sum\limits_{e\ni x}f(e)+\sum\limits_{y\in N(x)}f(y),$$ where $N(x)=\{y\in V(G)|xy\in…

We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…

The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components.…

A vertex colouring $f:V(G)\to C$ of a graph $G$ is complete if for any two distinct colours $c_1,c_2\in C$ there is an edge $\{v_1,v_2\}\in E(G)$ such that $f(v_i)=c_i$, $i=1,2$. The achromatic number of $G$ is the maximum number…

Let G be a group, and let M=(m_n) be a sequence of finitely supported probability measures on G. Consider the probability that two elements chosen independently according to m_n commute. Antolin, Martino and Ventura define the 'degree of…

We prove that every $n$-vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9-o(1)}$, while there exists an $n$-vertex tournament $G$ whose every acyclic subgraph has chromatic number at most $n^{3/4+o(1)}$.…

Ohba has conjectured \cite{ohb} that if the graph $G$ has $2\chi(G)+1$ or fewer vertices then the list chromatic number and chromatic number of $G$ are equal. In this paper we prove that this conjecture is asymptotically correct. More…

An adjacent vertex distinguishing edge colouring of a graph $G$ without isolated edges is its proper edge colouring such that no pair of adjacent vertices meets the same set of colours in $G$. We show that such colouring can be chosen from…

We show that Erd\H{o}s-R\'enyi random graphs $G(n,p)$ with constant density $p<1$ have correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger's conjecture for correspondence coloring. The proof…