The set chromatic number of a graph

Graphs and Combinatorics, 1990

The total chromatic number, Z"(G), of a graph G, is defined to be the minimum number ofcolours needed to colour the vertices and edges of a graph in such a way that no adjacent vertices, no adjacent edges and no incident vertex and edge are given the same colour. This paper shows that)('(G) _< z'(G) + 2x/~G), where z(G)is the vertex chromatic number and)((G)is the edge chromatic number of the graph.

New Trends in Mathematical Science, 2017

Graph coloring is one of the most important concept in graph theory. Many practical problems can be formulated as graph coloring problems. In this paper, we define a new coloring concept called local connective coloring. A local connective k-coloring of a graph G is a proper vertex coloring, which assigns colors from {1, 2, ..., k} to the vertices V (G) in a such way that any two nonadjacent vertices u and v of a color i satisfies κ(u, v) i, where κ(u, v) is the maximum number of internally disjoint paths between u and v. Adjacent vertices are colored with different colors as in the proper coloring. The smallest integer k for which there exists a local connective k-coloring of G is called the local connective chromatic number of G, and it is denoted by χ lc (G). We study this coloring on several classes of graphs and give some general bounds. We also compare local connective chromatic number of a graph with chromatic number and packing chromatic number of it.

Journal of Physics: Conference Series, 2021

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 ...

Journal of Discrete Mathematical Sciences and Cryptography, 2019

A k-coloring of a graph is a vertex coloring of G that uses k-colors. The chromatic number c(G) of a graph G is the minimum number of independent subsets that partition the vertex set of G. Any such minimum partition is called a chromatic partition of V(G). In this paper chromatic number of Sunlet and Bistar families of graphs are found

2001

Computing the chromatic number of a graph is an NP-hard problem. For random graphs and some other classes of graphs, estimators of the expected chromatic number have been well studied. In this paper, a new 0–1 integer programming formulation for the graph coloring problem is presented. The proposed new formulation is used to develop a method that generates graphs of known chromatic number by using the KKT optimality conditions of a related continuous nonlinear program.

European Journal of Combinatorics, 2007

2009

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such

Electronic Notes in Discrete Mathematics

A k−coloring of a graph G = (V, E) is a k-partition Π = {S 1 ,. .. , S k } of V into independent sets, called colors. A k-coloring is called neighbor-locating if for every pair of vertices u, v belonging to the same color S i , the set of colors of the neighborhood of u is different from the set of colors of the neighborhood of v. The neighbor-locating chromatic number, χ N L (G), is the minimum cardinality of a neighbor-locating coloring of G. In this paper, we examine the neighbor-locating chromatic number for various graph operations: the join, the disjoint union and Cartesian product. We also characterize all connected graphs of order n ≥ 3 with neighbor-locating chromatic number equal either to n or to n − 1 and determine the neighbor-locating chromatic number of split graphs.

Discrete Mathematics, 2012

A proper vertex k-coloring of a graph G is called dynamic, if there is no vertex v ∈ V (G) with d(v) ≥ 2 and all of its neighbors have the same color. The smallest integer k such that G has a k-dynamic coloring is called the dynamic chromatic number of G and denoted by χ 2 (G). We say that v ∈ V (G) in a proper vertex coloring of G is a bad vertex if d(v) ≥ 2 and only one color appears in the neighbors of v. In this paper, we show that if G is a graph with the chromatic number at least 6, then there exists a proper vertex χ (G)-coloring of G such that the set of bad vertices of G is an independent set. Also, we provide some upper bounds for χ 2 (G) − χ (G) in terms of some parameters of the graph G.

Discrete Applied Mathematics, 2011

A dynamic coloring of a graph G is a proper coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant c such that for every k-regular graph G, χ d (G) ≤ χ(G) + c ln k + 1. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.