Related papers: The digrundy number of digraphs
The arc graph $\delta(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta(G)$ join consecutive arcs of $G$. In 1981, Poljak and R\"{o}dl characterised the chromatic number of $\delta(G)$…
A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the…
The odd colouring number is a new graph parameter introduced by Petru\v{s}evski and \v{S}krekovski. In this note, we show that graphs with so called product structure have bounded odd-colouring number. By known results on the product…
In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $\gamma$ of a digraph $D$ is proper if no two arcs with the same head or with the same tail are assigned the same…
A well known problem from an excellent book of Lov\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh…
The chromatic number, which refers to the minimum number of colours required to colour the vertices of graphs properly, is one of the most central notions of the graph chromatic theory. Several of its aspects of interest have been…
By a $z$-coloring of a graph $G$ we mean any proper vertex coloring consisting of the color classes $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1 \leq i < j \leq k$, any vertex of color $j$ is adjacent to a…
In this paper, we study the domination number of middle graphs. Indeed, we obtain tight bounds for this number in terms of the order of the graph. We also compute the domination number of some families of graphs such as star graphs, double…
The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a…
A new algorithm to obtain the chromatic number of a finite, connected graph is proposed in this paper. The algorithm is based on contraction of non adjacent vertices.
The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le…
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The distinguishing chromatic number $\chi_{D}(G)$ of $G$ is…
This paper investigates when countable graphs have a finite or an infinite chromatic number through model theoretic methods. For Fra\"{i}ss\'{e} limits, we show that instability forces the chromatic number to be infinite, yielding a…
We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many…
We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum…
Nordhaus and Gaddum proved sharp upper and lower bounds on the sum and product of the chromatic number of a graph and its complement. Over the years, similar inequalities have been shown for a plenitude of different graph invariants. In…
While solving a question on list coloring of planar graphs, Dvo\v{r}\'{a}k and Postle introduced the new notion of DP-coloring (they called it correspondence coloring). A DP-coloring of a graph $G$ reduces the problem of finding a coloring…
There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic…
DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvo\v{r}\'{a}k and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic…
Brooks' Theorem states that a connected graph $G$ of maximum degree $\Delta$ has chromatic number at most $\Delta$, unless $G$ is an odd cycle or a complete graph. A result of Johansson (1996) shows that if $G$ is triangle-free, then the…