Related papers: Odd Chromatic Number of Graph Classes

We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…

Call a colouring of a graph \emph{distinguishing} if the only automorphism of this graph which preserves said colouring is the identity. Let $H$ be an arbitrary graph. We say that a graph $G$ is \emph{$H$-free} if $G$ does not contain an…

A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$,…

The defective chromatic number of a graph class is the infimum $k$ such that there exists an integer $d$ such that every graph in this class can be partitioned into at most $k$ induced subgraphs with maximum degree at most $d$. Finding the…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

A graph is called $k$-critical if its chromatic number is $k$ but any proper subgraph has chromatic number less than $k$. An old and important problem in graph theory asks to determine the maximum number of edges in an $n$-vertex…

A colouring of a graph $G$ has clustering $k$ if the maximum number of vertices in a monochromatic component equals $k$. Motivated by recent results showing that many natural graph classes are subgraphs of the strong product of a graph with…

An edge-coloring of a graph $G$ with consecutive integers $c_{1},\ldots,c_{t}$ is called an \emph{interval $t$-coloring} if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of…

We define the $d$-defective incidence chromatic number of a graph, generalizing the notion of incidence chromatic number, and determine it for some classes of graphs including trees, complete bipartite graphs, complete graphs, and…

Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ with $\Delta(G)>|V(G)|/3$ has chromatic…

Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a graph $G$ satisfies $\Gamma(G)\ge t$ if and only if $G$ contains an induced subgraph called a $t$-atom.The family of $t$-atoms has bounded order and contains a finite…

Given a hypergraph $G$ and a subhypergraph $H$ of $G$, the \emph{odd Ramsey number} $r_{odd}(G,H)$ is the minimum number of colors needed to edge-color $G$ so that every copy of $H$ intersects some color class in an odd number of edges.…

Let $\chi'_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 \pmod k$. Scott [{\em Discrete Math. 175}, 1-3…

The dichromatic number of a digraph is the minimum integer $k$ such that it admits a $k$-dicolouring, i.e. a partition of its vertices into $k$ acyclic subdigraphs. We say that a digraph $D$ is a super-orientation of an undirected graph $G$…

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ that $G$ has a $k-$injective coloring is called injective chromatic number of $G$ and denoted by…

As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every…

Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P.…

A graph $G$ is $(1,3)$-colorable if its vertices can be partitioned into subsets $V_1$ and $V_2$ so that every vertex in $G[V_1]$ has degree at most $1$ and every vertex in $G[V_2]$ has degree at most $3$. We prove that every graph with…