Colouring graphs with bounded generalized colouring number

Colouring edges with many colours in cycles

A graph in this paper is nontrivial, finite, connected, simple, and undirected. Graph consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of is not equal with the weight of , where the weight of (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where is the colour on the vertex . The vertex local antimagic chromatic number is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Let 𝐺 be a connected graph and П={𝑆1,𝑆2,…,𝑆𝑘} be an ordered partition of 𝑉(𝐺). Let 𝑆𝑖 is a set of color classes using colors 1,2,...,𝑘 where 𝑘 as positive integer. The color code 𝑐П(𝑣) of vertex 𝑣 in 𝐺 with respect to П is defined as 𝑘-vector, 𝑐П(𝑣)=(𝑑(𝑣,𝑆1),𝑑(𝑣,𝑆2),…,𝑑(𝑣,𝑆𝑖)) where 𝑑(𝑣,𝑆𝑖)=𝑚𝑖𝑛{𝑑(𝑣,𝑥)|𝑥∈𝑆𝑖} for 1≤𝑖≤𝑘. If each of vertices in 𝐺 have distinct color codes, then 𝑐 is called as locating coloring of 𝐺. The minimum number of colors that are used for locating coloring is called as locating chromatic number of 𝐺, denoted by 𝜒𝐿(𝐺). In this article we will discuss the chromatic number of the side amalgamation location of the circle graph 𝑎𝑚𝑎𝑙𝑠(𝐶𝑛𝑗;𝑣𝑗,𝑙𝑣𝑗,𝑛) with 𝑛=3,4,1≤𝑗≤𝑚, and 𝑚≥2.

Minimal coloring and strength of graphs

Extending precolorings to circular colorings

Universal [formula omitted]-colorable graphs without a given configuration

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.

Fix positive integers p and q with 2 ⩽ q ⩽ ( p 2 ) . An edge coloring of the complete graph K n is said to be a ( p , q ) -coloring if every K p receives at least q different colors. The function f ( n , p , q ) is the minimum number of colors that are needed for K n to have a ( p , q ) -coloring. This function was introduced about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that f ( n , p , p ) is polynomial in n and asked to determine the maximum q, depending on p, for which f ( n , p , q ) is subpolynomial in n. We prove that the answer is p − 1 .

A polyhedral study of the acyclic coloring problem

Minimum sum coloring problem: Upper bounds for the chromatic strength

Incidence coloring of Mycielskians with fast algorithm

Acyclic and k-distance coloring of the grid

The classical Nelson — Erdős — Hadwiger problem, which was proposed in the late 1940’s (see [1], [2]), consists in finding or at least estimating the so-called chromatic number χ(ℝ d ) of the Euclidean space ℝ d , where the value χ(ℝ d ) is defined as the minimum number of colours needed to paint all the points in ℝ d in such a way that any two points at the distance 1 apart receive different colours. Of course, in this definition, distance 1 may be replaced by an arbitrary fixed distance a > 0, due to the homogeneity of the Euclidean space. In other words, the quantity χ(ℝ d ) is the usual chromatic number of a graph \(\mathfrak{G}_a^d = \left( {{\mathfrak{G}^d},\mathfrak{E}_a^d} \right)\), provided $${\mathfrak{V}^d} = {\mathbb{R}^d},\mathfrak{E}_a^d = \{ (a,y) \in {\mathfrak{V}^d} \times {\mathfrak{V}^d}:|a - y| = a\} ,\,a > 0$$ .

On f-colorings of nearly bipartite graphs

On the chromatic index of multigraphs and a conjecture of Seymour (I)