Thue-like Sequences and Rainbow Arithmetic Progressions

Colouring edges with many colours in cycles

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

Minimum sum coloring problem: Upper bounds for the chromatic strength

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 $${\mathfrak{V}^d} = {\mathbb{R}^d},\mathfrak{E}_a^d = \{ (a,y) \in {\mathfrak{V}^d} \times {\mathfrak{V}^d}:|a - y| = a\} ,\,a > 0$$ .

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.

Colouring graphs with bounded generalized colouring number

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.

Minimal coloring and strength of graphs

Recent efforts in our laboratory have been focused on the or­ganization of repetitive and nonrepetitive sequences in animal DNA, and the representation of these sequences in RNA. A new chapter in these studies began recently with observations that lead to the conclusion that middle-repetitive DNA sequences are interspersed in an orderly way amongst nonrepetitive sequence elements. I will begin by summarizing some of these recent experiments on sequence organization in the DNA’s of Xenopus and the sea urchin, and then proceed to describe some new findings on the sequence constitution of sea urchin messenger RNA.

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices have distinct sets of colors. The minimum number of colors needed for an adjacent vertex distinguishing total coloring of G is denoted by $\chi''_{a}(G)$ . In this paper, we characterize completely the adjacent vertex distinguishing total chromatic number of planar graphs G with large maximum degree Δ by showing that if Δ?14, then $\varDelta+1\leq \chi''_{a}(G)\leq \varDelta+2$ , and $\chi''_{a}(G)=\varDelta+2$ if and only if G contains two adjacent vertices of maximum degree.

For a simple connected graph G, let c : V(G) → ℕ be a vertex coloring of G, where adjacent vertices may be colored the same. The neighborhood color set of a vertex v, denoted by NC(v), is the set of colors of the neighbors of v. The coloring c is called a set coloring provided that NC(u) ≠ NC(v) for every pair of adjacent vertices u and v of G. The minimum number of colors needed for a set coloring of G is referred to as the set chromatic number of G and is denoted by χs(G). In this work, the set chromatic number of graphs is studied in relation to the graph operation called middle graph. Our results include the exact set chromatic numbers of the middle graph of cycles, paths, star graphs, double-star graphs, and some trees of height 2. Moreover, we establish the sharpness of some bounds on the set chromatic number of general graphs obtained using this operation. Finally, we develop an algorithm for constructing an optimal set coloring of the middle graph of trees of height 2 under some assumptions.

A proper k-edge coloring of a graph G is called adjacent vertex distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the color set of edges incident to u is not equal to the color set of edges incident to υ, where uυ ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by χ′aa(G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. In this paper we prove that if G(V, E) is a graph with no isolated edges, then χ′aa(G) ≤ 32Δ.

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

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )} is considered, for f ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .