Pseudoachromatic Number Research Articles

Computation of Pseudocoloring of Graphs through Python

In the task of coloring the vertices of a simple graph G one come across innumerable number of challenges. There are various graph coloring parameters available in the literature. The concept of pseudo coloring is quite interesting. In this type of coloring, we can allot same color to the adjacent vertices. The maximum number of colors used in a pseudocoloring where for any two distinct colors, one can always find at least one edge between them is called pseudo achromatic number, ѱ(G) of G. In the paper we determine this coloring parameter for some classes of graphs through python code.

The Hadwiger number, chordal graphs and [formula omitted]-perfection

A graph is chordal if every induced cycle has three vertices. The Hadwiger number is the order of the largest complete minor of a graph. We characterize the chordal graphs in terms of the Hadwiger number and we also characterize the families of graphs such that for each induced subgraph H, (1) the Hadwiger number of H is equal to the maximum clique order of H, (2) the Hadwiger number of H is equal to the achromatic number of H, (3) the b-chromatic number is equal to the pseudoachromatic number, (4) the pseudo-b-chromatic number is equal to the pseudoachromatic number, (5) the Hadwiger number of H is equal to the Grundy number of H, and (6) the b-chromatic number is equal to the pseudo-Grundy number.

Pseudo-Achromatic Number of Honeycomb Derived Networks

The Pseudo-Achromatic number ψs G of a graph G is the maximum number of colors which may be assigned to the vertices of G so that for every two colors, there exists adjacent vertices to which these colors are assigned (adjacent vertices may have the same color). This paper formulates an O(1)-approximation algorithm for the Pseudo-Achromatic number of honeycomb derived networks.

The ωψ-perfection of graphs

In this paper we study a natural generalization for the perfection of graphs to other interesting parameters related with colorations. This generalization was introduced partially by Christen and Selkow in 1979 and Yegnanarayanan in 2001.Let a,b∈{ω,χ,Γ,α,ψ} where ω is the clique number, χ is the chromatic number, Γ is the Grundy number, α is the achromatic number and ψ is the pseudoachromatic number. A graph G is ab-perfect if for every induced subgraph H, a(H)=b(H). In this work we characterize the ωψ-perfect graphs.

On the pseudoachromatic index of the complete graph

Let q = 2β be, for some β∈ℕ, and let n = q2 + q+ 1. By exhibiting a complete coloring of the edges of Kn, we show that the pseudoachromatic number ψ(Gn) of the complete line graph Gn = L(Kn)—or the pseudoachromatic index of Kn, if you will—is at least q3 + q. This bound improves the implicit bound of Jamison [Discrete Math 74 (1989), 99–115] which is given in terms of the achromatic number: ψ(Gn)≥α(Gn)≥q3 + 1. We also calculate, precisely, the pseudoachromatic number when q+ 1 extra points are added: Copyright © 2010 John Wiley Periodicals, Inc. J Graph Theory 66:89-97, 2011

On the pseudo-achromatic number problem

We study the parameterized complexity of the pseudo-achromatic number problem: Given an undirected graph and a parameter k , determine if the graph can be partitioned into k groups such that every two groups are connected by at least one edge. This problem has been extensively studied in graph theory and combinatorial optimization. We show that the problem has a kernel of at most ( k − 2 ) ( k + 1 ) vertices that is constructable in time O ( m n ) , where n and m are the number of vertices and edges, respectively, in the graph, and k is the parameter. This directly implies that the problem is fixed-parameter tractable. We also study generalizations of the problem and show that they are parameterized intractable.

Bounding the pseudoachromatic index of the complete graph via projective planes

Abstract Let q = 2 β and n = q 2 + q + 1 . Further, let G = L ( K n ) be the complete line graph and ψ ( G ) its pseudoachromatic number. By exhibiting an explicit colouring of E ( K n ) , we show that ψ ( G ) ⩾ q 3 + q . This result improves the bound ψ ( G ) ⩾ q 3 + 1 due to Jamison (1989) [Jamison, R.E.; On the edge achromatic numbers of complete graphs, Discrete. Math. 74 (1989) 99–115].

On the pseudoachromatic number of join of graphs

Given an undirected graph G = (V, E), with vertex set V and edge set E, the pseudoachromatic number ψ(G) of the graph G is the maximum number of colors used to color the vertices in such a way that, for any given pair of colors i, j there exists an edge e = (u, v) ∈ E(G) such that u is colored i and v is colored j. In this paper we give a complete characterization of when the ψ of the join of any two graphs is the sum of the ψ of the two graphs.

Graph colourings and partitions

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 briefly surveyed. Some new results were found along these directions. We have redefined the concept of perfect graphs in terms of these parameters and obtained a few results. Some open problems are raised.

Achromatic number versus pseudoachromatic number: a counterexample to a conjecture of Hedetniemi

The pseudoachromatic number of a graph is the largest number of colours in a (not necessarily proper) vertex colouring of the graph such that every pair of distinct colours appears on the endpoints of some edge. The achromatic number is largest number of colours which can be used if the colouring must also be proper. Hedetniemi (http://cyclone.cs.clemson.edu/ ̃hedet/coloring.html) conjectured that these two parameters are equal for all trees. We disprove this conjecture by giving an infinite family of trees for which the pseudoachromatic number strictly exceeds the achromatic number.

On the existence of graphs with prescribed coloring parameters

The pseudoachromatic number ψ(G) of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that between any two distinct parts, there is at least one edge of G. Here, we prove that if 2⩽a⩽b⩽c, then there exists a graph G with chromatic number a, achromatic number b and pseudoachromatic number c.

The Pseudoachromatic Number of a Graph

The Pseudoachromatic Number of a Graph

The Pseudoachromatic Number of a Graph

The pseudoachromatic number of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that, between any two distinct parts, there is at least one edge of G. This parameter is determined for graphs such as cycles, paths, wheels, certain complete multipartite graphs, and for other classes of graphs. Some open problems are raised.

Extremal graphs in some coloring problems

For a simple graph G with chromatic number χ( G), the Nordhaus-Gaddum inequalities give upper and lower bounds for χ( G) χ( G c ) and χ( G) + χ( G c ). Based on a characterization by Fink of the extremal graphs G attaining the lower bounds for the product and sum, we characterize the extremal graphs G for which A( G) B( G c ) is minimum, where A and B are each of chromatic number, achromatic number and pseudoachromatic number. Characterizations are also provided for several cases in which A( G) + B( G c ) is minimum.

On the pseudoachromatic number of a graph

On the pseudoachromatic number of a graph

Partition graphs and coloring numbers of a graph

In [2] and [3] the achromatic number of a graph was introduced. Several bounds for this parameter were obtained by Harary and Hedetmem and Gupta. In [4] Gupta introduced the pseudoachromatic number of a graph and obtained some bounds for this number. In this note we obtain several other bounds for these numbers and generalize some of the known results using the concept of a “partition graph” of a graph.