Chromatic Partition Research Articles

Topological structures induced by chromatic partitioning of vertex set of graphs

This paper presents a method of constructing topologies on vertex set of a graph G induced by chromatic partition of vertex set of the graph. It introduces colour lower approximation and colour upper approximation of vertex induced subgraphs and acquaints the open and closed sets of the topology generated by chromatic partition on the vertex set of graphs. It explores some of the properties of colour lower approximation and colour upper approximation of vertex induced subgraphs. It also establishes some new subgraphs based on the colour lower approximation and colour upper approximation and some of their properties have been studied.

A heuristic for the minimum cost chromatic partition problem

The graph coloring problem consists in coloring the vertices of a graphG=(V, E)with a minimum number of colors, such as that any two adjacent vertices receive different colors. The minimum cost chromatic partition problem (MCCPP) is an extension of the graph coloring problem in which there are costs associated with the colors and one seeks a vertex coloring minimizing the sum of the costs of the colors used in each vertex. The problem finds applications in VLSI design and in some scheduling problems modeled on interval graphs. We propose a trajectory search heuristic using local search, path-relinking, and perturbations for solving MCCPP and discuss computational results.

A note on polynomial algorithm for cost coloring of bipartite graphs with Δ ≤ 4

In the note we consider vertex coloring of a graph in which each color has an associated cost which is incurred each time the color is assigned to a vertex. The cost of coloring is the sum of costs incurred at each vertex. We show that the minimum cost coloring problem for n-vertex bipartite graph of degree ∆≤4 can be solved in O(n^2) time. This extends Jansen’s result [K.Jansen,The optimum cost chromatic partition problem, in: Proc. CIAC’97,Lecture Notes in Comput. Sci. 1203 (1997) 25–36] for paths and cycles to subgraphs of biquartic graphs.

Chromatic number of some families of graphs

A k-coloring of a graph is a vertex coloring of G that uses k-colors. The chromatic number χ(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

Computing lower bounds for minimum sum coloring and optimum cost chromatic partition

The Minimum Sum Coloring Problem (MSCP) and Optimum Cost Chromatic Partition Problem (OCCP), variants of the well-known Graph Coloring Problem (GCP), find applications in different domains, such as VLSI design, resource allocation, scheduling, and so on. MSCP and OCCP are much harder than GCP and solving them for large graphs is particularly challenging. In the literature, much effort has been spent to develop upper and lower bounds for MSCP. Lower bounds for MSCP are not only interesting in theory but also useful in practice: lower bounds can be used to accelerate solvers for MSCP and evaluate the quality of heuristic results. In this paper, we propose two new theoretical lower bounds for MSCP and OCCP by exploiting structural properties of them. The two lower bounds are based on (relaxed) values of the chromatic number, independence number and bipartite number of the graph. Experiments on standard benchmarks DIMACS and COLOR show that our lower bounds can improve previous known theoretical and computational lower bounds for MSCP on at least 22% of the benchmark instances.

Planar and Non Planar Construction of - Uniquely Colorable Graph

A uniquely colorable graph G whose chromatic partition contains atleast one g - set is termed as a g - uniquely colorable graph. In this paper, we provide necessary and sufficient condition for and G* to be g - uniquely colorable whenever G g- uniquely colorable and also provide constructive characterization to show that whenever G is g- uniquely colorable such that |P | ³ 2, G can be both planarand non planar.

Planar graph characterization of γ - Uniquely colorable graphs

A graph G = ( V, E ) is uniquely colorable if the chromatic number χ ( G ) = n and every n – coloring of G induces the same partition of V. Uniquely colorable graphs whose chromatic partition contains at least one γ - set is γ - uniquely colorable graphs In this paper, we characterize the planarity of γ - uniquely colorable graphs.

Minimum sum coloring problem: Upper bounds for the chromatic strength

The minimum sum coloring problem (MSCP) is a vertex coloring problem in which a weight is associated with each color. Its aim is to find a coloring of the vertices of a graph G with the minimum sum of the weights of the used colors. The MSCP has important applications in the fields such as scheduling and VLSI design. The minimum number of colors among all optimal solutions of the MSCP for G is called the chromatic strength of G and is denoted by s(G). A tight upper bound of s(G) allows to significantly reduce the search space when solving the MSCP. In this paper, we propose and empirically evaluate two new upper bounds of s(G) for general graphs, UBA and UBS, based on an abstraction of all possible colorings of G formulated as an ordered set of decreasing positive integer sequences. The experimental results on the standard benchmarks DIMACS and COLOR show that UBA is competitive, and that UBS is significantly tighter than those previously proposed in the literature for 70 graphs out of 74 and, in particular, reaches optimality for 8 graphs. Moreover, both UBA and UBS can be applied to the more general optimum cost chromatic partition (OCCP) problem.

Tight Just Chromatic Excellence In Fuzzy Graphs

Let G be a simple fuzzy graph. A family  I“a¶ = { I³1, I³2,…, I³k} of fuzzy sets on a set V is called k-fuzzy colouring of V = (V,Iƒ,µ) if i) âˆa I“a¶ = Iƒ, ii) I³i∩ I³j = Ф, iii) for every strong edge (x,y) (i.e., µ(xy) > 0) of G min{I³i(x), I³j(y)} = 0; (1 ≤ i ≤ k). The minimum number of k for which there exists a k-fuzzy colouring is called the fuzzy chromatic number of G denoted as I‡f (G). Then I“a¶  is the partition of independent sets of vertices of G in which each sets has the same colour is called the fuzzy chromatic partition. A graph G is called the just I‡f -excellent if every vertex of G appears as a singleton in exactly one _f -partition of G. A just I‡f –excellent graph of order n is called the tight just I‡f -excellent if G having exactly n, I‡f -partitions. This paper aims at the study of the new concept namely tight just Chromatic excellence in fuzzy graphs and its properties. 02000 Mathematics Subject Classification:05C72 Key words: fuzzy just chromatic excellent, tight just I‡f -excellent, fuzzy colourful vertex, fuzzy kneser graph.

Chromatic Excellence in Graphs

Excellence in graphs introduced by G.H. Fricke is extended to partitions of the vertex set with respect to a parameter. A graph G is said to be Chromatic excellent if {v} appears in a chromatic partition of G for every vϵV(G). This paper is devoted to the study of chromatic excellence in graphs.

Hamiltonian Partition Coloring

Coloring of vertices/edges in graphs has been studied by many authors and various types of coloring were also introduced. A proper coloring is a partition of the vertex set into independent sets based on the properties of the chromatic partition ( which is a partition of minimum cardinality into independent sets). Other notions like achromatic coloring, b-coloring etc, have been introduced. A proper coloring P of G is called a hamiltonian partition coloring if P ( G ) is hamiltonian. The maximum cardinality of a hamiltonian partition coloring of G is called the hamiltonian partition achromatic number of G and is denoted by h ( G ). The hamiltonian coloring of a connected graph G introduced by Chartrand et al [1] is different from hamiltonian partition coloring. In this paper, we characterize graphs which has a hamiltonian partition. Also, we give example of graphs having prescribed chromatic numbers and hamiltonian partition numbers. We derive results connecting the hamiltonian chromatic number of G 1 [ G 2 and G 1 + G 2 .

Differential Coloring of Graphs

Let G=(V,E) be a connected graph. For a set S⊆V, the differential of S, denoted by ∂(S), is defined to be ∂(S)=| B(S) |−| S |, where B(S)=N(S)∩(V−S). A set S⊆V is said to be a positive differential set if ∂(S)≥0. A partition {V1,V2,…,Vk} of V is said to be a positive differential chromatic partition of G if each Vi is both independent and positive differential. The minimum order of a positive differential chromatic partition of G is called the differential chromatic number of G and is denoted by χ∂(G). In this paper we initiate a study of this parameter.

The chromatic sum of a graph: history and recent developments

The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength of a graph is the minimum number of colors necessary to obtain its chromatic sum. A natural generalization of chromatic sum is optimum cost chromatic partition (OCCP) problem, where the costs of colors can be arbitrary positive numbers. Existing results about chromatic sum, strength of a graph, and OCCP problem are presented together with some recent developments. The focus is on polynomial algorithms for some families of graphs and NP‐completeness issues.

Approximation Results for the Optimum Cost Chromatic Partition Problem

In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring cost is minimum. We prove several approximation results for the OCCP problem restricted to bipartite, chordal, comparability, interval, permutation, split, and unimodular graphs. We prove that there exists no polynomial approximation algorithm with ratio O(|V|0.5−ϵ) for the OCCP problem restricted to bipartite and interval graphs, unless P=NP. Furthermore, we propose approximation algorithms with ratio O(|V|0.5) for bipartite, interval, and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O(|V|1−ϵ) for the OCCP problem restricted to split, chordal, permutation, and comparability graphs, unless P=NP.

A chromatic partition polynomial

A polynomial in two variables is defined by C n ( x , t ) = Σ π ∈ Π n x ( G π , x ) t | π | , where Π n is the lattice of partitions of the set {1, 2, …, n}, G π is a certain interval graph defined in terms of the partition gp, χ( G π, x ) is the chromatic polynomial of G π and |π| is the number of blocks in π. It is shown that C n ( x , t ) = Σ k = 0 n t k Σ i = 0 k ( n − k n − 1 ) S ( n , i ) ( x ) i , where S( n, i) is the Stirling number of the second kind and ( x) i = x( x − 1) ··· ( x − i + 1). As a special case, C n (−1, − t) = A n ( t), where A n ( t) is the nth Eulerian polynomial. Moreover, A n ( t ) = Σ π ∈ Π n a π t | π | where a π is the number of acyclic orientations of G π .

Colorability of induced matroids

Let G be a bipartite graph with bipartition { X, Y} such that there is a matching which matches Y with a subset of X. Let M be a matroid on X, and for each positive integer k, let M ( k) be the matroid on X which is the union of M with itself k times. The G-induced chromatic number of M is the smallest integer p such that Y is an independent set of the induced matroid ( M ( p) ) G . This chromatic number gives rise to a partition of | Y| called the G-induced chromatic partition. This partition majorizes partitions corresponding to other induced partitions, of Y.

The Generalisation of Tutte's Result for Chromatic Trees, by Lagrangian Methods

A K-coloured rooted tree t is said to have colour partitionL if L is a K × ∞ matrix with elements lij equal to the number of non-root vertices of colour i and degree j. If adjacent vertices are of different colours then t is called a chromatic tree and L a chromatic partition. The tree has edge partitionD where D is a K × K matrix with elements dij equal to the number of edges, directed away from the root, from a vertex of colour i to a vertex of colour j.In this paper we consider a method for enumerating trees with respect to colour and degree information. The method makes use of elementary decompositions of trees, and the functional equations which are induced. A number of new results are obtained by this means. More specifically, we consider (Section 3) the enumeration of rooted plane X-coloured trees with given colour and edge partitions.

A new method to find out the chromatic partition of a symmetric graph

In this paper an algorithm to find out the chromatic partition of a symmetric graph is described. The algorithm is developed in two stages. In the first stage, all possible maximal independent sets of vertices of the graph ore determined starting from adjacency matrix description of the given graph. Several important results are arrived at about the lower bound on the chromatic number of the graph. Lastly, in the second phase of the algorithm, a minimal cover of the vertices with the maximal independent seta is enumerated by introducing the classical concept of prime implicant covering to obtain the desired chromatic partition of the graph. The proposed algorithm is simple and straightforward and may be programmed on a general purpose digital computer.

A practical approach to the chromatic partition problem

A new approach is presented for finding a chromatic partition of the nodes of an arbitrary, finite, connected, linear graph. Based on this approach, three algorithms are presented which require much less enumeration than conventional procedures. In fact, hand computation time is reduced from hours to minutes.