Colouring edges with many colours in cycles

List strong edge-coloring of graphs with maximum degree 4

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

Star Chromatic Index of Subcubic Graphs

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph. Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

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 .

Minimum sum coloring problem: Upper bounds for the chromatic strength

A polyhedral study of the acyclic coloring problem

A tree in an edge-colored connected graph G is called a rainbow tree if no two edges of it are assigned the same color. For a vertex subset $$S\subseteq V(G)$$ , a tree is called an S-tree if it connects S in G. A k-rainbow coloring of G is an edge-coloring of G having the property that for every set S of k vertices of G, there exists a rainbow S-tree in G. The minimum number of colors that are needed in a k-rainbow coloring of G is the k-rainbow index of G, denoted by $${\mathrm {rx}}_k(G)$$ . The Steiner distance d(S) of a set S of vertices of G is the minimum size of an S-tree T. The k-Steiner diameter $${\mathrm {sdiam}}_k(G)$$ of G is defined as the maximum Steiner distance of S among all sets S with k vertices of G. In this paper, we focus on the 3-rainbow index of graphs and find all finite families $$\mathcal {F}$$ of connected graphs, for which there is a constant $$C_\mathcal {F}$$ such that, for every connected $$\mathcal {F}$$ -free graph G, $${\mathrm {rx}}_3(G)\le {\mathrm {sdiam}}_3(G)+C_\mathcal {F}$$ .

A bipartite graph is Hamiltonian laceable if any two of its vertices in different partite sets are connected by a Hamiltonian path. A Hamiltonian laceable graph $G$ is called strongly Hamiltonian laceable if any two of its vertices in the same partite set are connected by a path of length $|V(G)|-2$. A Hamiltonian laceable graph $G$ (with two partite sets $V_0, V_1$) is called hyper-Hamiltonian laceable, if for any vertex $v \in V_{i}$ for $i \in \{0,1\}$, there is a Hamiltonian path of $G-\{v\}$ between any two vertices in $V_{1-i}$. In this paper, we focus on the edge-fault-tolerant strongly Hamiltonian laceability and hyper-Hamiltonian laceability on the class of Cayley graphs generated by transposition trees, which are a generalization of star graph and bubble-sort graph. For every $n$-dimensional Cayley graph generated by a transposition tree $\Gamma _n$, we show that $\Gamma _{n}-F$ is strongly Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-3$, which generalizes results in [ 1, 11], and show that $\Gamma _{n}-F$ is hyper-Hamiltonian laceable for any $F \subseteq E(\Gamma _{n})$ with $|F|\leq n-4$.

Long paths in hypercubes with conditional node-faults

The 2-path-bipanconnectivity of hypercubes