EQUIVALENCE COLOURING OF GRAPHS (original) (raw)

2001

In this paper we have investigated mainly the three colouring parameters of a graph G, viz., the chromatic number, the achromatic number and the pseudoachromatic number. The importance of their study in connection with the computational complexity, partitions, algebra, projective plane geometry and analysis were brie y surveyed. Some new results were found along these directions. We have redeÿned the concept of perfect graphs in terms of these parameters and obtained a few results. Some open problems are raised.

New results on generalized graph coloring

2004

For graph classes P 1 , . . . , P k , Generalized Graph Coloring is the problem of deciding whether the vertex set of a given graph G can be partitioned into subsets V 1 , . . . ,V k so that V j induces a graph in the class P j ( j = 1, 2, . . . , k). If P 1 = · · · = P k is the class of edgeless graphs, then this problem coincides with the standard vertex k-COLORABILITY, which is known to be NP-complete for any k ≥ 3. Recently, this result has been generalized by showing that if all P i 's are additive hereditary, then the generalized graph coloring is NP-hard, with the only exception of bipartite graphs.

On Pseudocomplete and Complete Coloring of Graphs

2013

A complete partition of a graph G = (V, E) is a partition of the vertex set V in which there is an edge connecting every pair of distinct classes. That is, a partition V 1 ,…,V t of V is complete if, for every i, j, i ≠ j, there is an edge {v i , v j } such that v i ∈ V i and v j ∈ V j. In this paper we study the problem of finding a complete partition of the largest possible size for certain finite, simple and undirected graphs. This concept is interpreted in terms of graph vertex colorings. In the process we give (1)illustrations, (2)prove results, (3)indicate the scope of its application and also raise some open problems.

A note on selective line-graphs and partition colorings

Operations Research Letters, 2019

We extend the definition of sandwich line-graphs, a class of auxiliary graphs the stable sets of which are in 1-to-1 correspondence with the colorings of the original graph, from graphs to partitioned graphs, this way, we obtain a one-to-one correspondence between stable sets and partition colorings.

On the total and AVD-total coloring of graphs

AKCE International Journal of Graphs and Combinatorics, 2020

A total coloring of a graph G is an assignment of colors to the vertices and the edges such that (i) no two adjacent vertices receive same color, (ii) no two adjacent edges receive same color, and (iii) if an edge e is incident on a vertex v, then v and e receive different colors. The least number of colors sufficient for a total coloring of graph G is called its total chromatic number and denoted by v 00 ðGÞ: An adjacent vertex distinguishing (AVD)-total coloring of G is a total coloring with the additional property that for any adjacent vertices u and v, the set of colors used on the edges incident on u including the color of u is different from the set of colors used on the edges incident on v including the color of v. The adjacent vertex distinguishing (AVD)-total chromatic number of G, v 00 a ðGÞ is the minimum number of colors required for a valid AVD-total coloring of G. It is conjectured that v 00 ðGÞ DðGÞ þ 2, which is known as total coloring conjecture and is one of the famous open problems. A graph for which the total coloring conjecture holds is called totally colorable graph. The problem of deciding whether v 00 ðGÞ ¼ DðGÞ þ 1 or v 00 ðG ¼ DðGÞ þ 2 for a totally colorable graph G is called the classification problem for total coloring. However, this classification problem is known to be NP-hard even for bipartite graphs. In this paper, we give a sufficient condition for a bipartite biconvex graph G to have v 00 ðGÞ ¼ DðGÞ þ 1: Also, we propose a linear time algorithm to compute the total chromatic number of chain graphs, a proper subclass of biconvex graphs. We prove that the total coloring conjecture holds for the central graph of any graph. Finally, we obtain the AVD-total chromatic number of central graphs for basic graphs such as paths, cycles, stars and complete graphs.

Equitable colorings of Cartesian products of graphs

Discrete Applied Mathematics, 2012

The present paper studies the following variation of vertex coloring on graphs. A graph G is equitably k-colorable if there is a mapping f : V (G) → {1, 2,. .. , k} such that f (x) ̸ = f (y) for xy ∈ E(G) and || f −1 (i) | − | f −1 (j) ||≤ 1 for 1 ≤ i, j ≤ k. The equitable chromatic number of a graph G, denoted by χ = (G), is the minimum k such that G is equitably k-colorable. The equitable chromatic threshold of a graph G, denoted by χ * = (G), is the minimum t such that G is equitably k-colorable for all k ≥ t. Our focus is on the equitable colorability of Cartesian products of graphs. In particular, we give exact values or upper bounds of χ = (G H) and χ * = (G H) when G and H are cycles, paths, stars, or complete bipartite graphs.

Selective line-graphs and partition colorings

2019

Neighbor-locating coloring: graph operations and extremal cardinalities

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.

Selective graph coloring in some special classes of graphs

In this paper, we consider the selective graph coloring problem. Given an integer k ≥ 1 and a graph G = (V, E) with a partition V1, . . . , Vp of V , it consists in deciding whether there exists a set V * in G such that |V * ∩ Vi| = 1 for all i ∈ {1, . . . , p}, and such that the graph induced by V * is k-colorable. We investigate the complexity status of this problem in various classes of graphs.

EQUIVALENCE COLOURING OF GRAPHS (original) (raw)

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