A note on vertex partitions

Lecture Notes in Computer Science, 1999

Let an edge cut partition the vertex set of a graph into k disjoint subsets A1, . . . , A k with ||Ai| − |Aj|| ≤ 1. We consider the problem of determining the minimal size of such a cut for a given graph. For this we introduce a new lower bound method which is based on the solution of an extremal set problem and present bounds for some graph classes based on Hamming graphs.

Discrete Mathematics, 2010

Cographs form the minimal family of graphs containing K 1 that is closed with respect to complementation and disjoint union. We discuss vertex partitions of graphs into the smallest number of cographs. We introduce a new parameter, calling the minimum order of such a partition the c-chromatic number of the graph. We begin by axiomatizing several well-known graphical parameters as motivation for this function. We present several bounds on c-chromatic number in terms of well-known expressions. We show that if a graph is triangle-free, then its chromatic number is bounded between the c-chromatic number and twice this number. We show that both bounds are sharp for graphs with arbitrarily high girth. This provides an alternative proof to a result by Broere and Mynhardt; namely, there exist triangle-free graphs with arbitrarily large c-chromatic numbers. We show that any planar graph with girth at least 11 has a c-chromatic number at most two. We close with several remarks on computational complexity. In particular, we show that computing the c-chromatic number is NP-complete for planar graphs.

Information Processing Letters, 1988

The Uniform Partition (UP) and the Simple Max Partition (SMP) problems arc NP-complete graph partitioning problems, and polynomial-time algorithms are known for few classes of graphs only. In the present paper, the class of line-graphs is considered and a polynomial algorithm is proposed to solve both problems in this class. When the instance space is extended to digraphs, a characterization is possible which leads to similar results.

Journal of Graph Theory, 2015

Given k ≥ 1, a k-proper partition of a graph G is a partition P of V (G) such that each part P of P induces a k-connected subgraph of G. We prove that if G is a graph of order n such that δ(G) ≥ √ n, then G has a 2-proper partition with at most n/δ(G) parts. The bounds on the number of parts and the minimum degree are both best possible. We then prove that If G is a graph of order n with minimum degree δ(G) ≥ c(k − 1)n, where c = 2123 180 , then G has a k-proper partition into at most cn δ(G) parts. This improves a result of Ferrara, Magnant and Wenger [Conditions for Families of Disjoint k-connected Subgraphs in a Graph, Discrete Math. 313 (2013), 760-764] and both the degree condition and the number of parts are best possible up to the constant c.

Combinatorics, Probability and Computing, 1998

We consider the function χ(Gk), defined to be the smallest number of colours that can colour a graph G in such a way that no vertices of distance at most k receive the same colour. In particular we shall look at how small a value this function can take in terms of the order and diameter of G. We get general bounds for this and tight bounds for the cases k=2 and k=3.

Journal of Mathematical Sciences, 2004

Discrete Mathematics, 2009

Given integers j and k and a graph G, we consider partitions of the vertex set of G into j + k parts where j of these parts induce empty graphs and the remaining k induce cliques. If such a partition exists, we say G is a (j, k)-graph. For a fixed j and k we consider the maximum order n where every graph of order n is a (j, k)-graph. The split-chromatic number of G is the minimum j where G is a (j, j)-graph. Further, the cochromatic number is the minimum j+k where G is a (j, k)-graph. We examine some relations between cochromatic, split-chromatic and chromatic numbers. We also consider some computational questions related to chordal graphs and cographs.

Theoretical Computer Science, 2020

As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 ,. .. , S k ∈ Π such that d(u, S i) = d(v, S i) for every i ∈ {1,. .. , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1,. .. , r}.

Procedia Computer Science, 2015

The concept of a graph partition dimension was introduced by Chartrand et al. (1998). Let Π = {L 1 , L 2 , L 3 , • • • , L k } be a k-partition of V(G). The representation r(v|Π) of a vertex v with respect to Π is the vector (d(v, L 1), d(v, L 2), • • • , d(v, L k)). The partition Π is called a resolving partition of G if r(w|Π) r(v|Π) for all distinct w, v ∈ V(G). The partition dimension of a graph, denoted by pd(G), is the cardinality of a minimum resolving partition of G. This paper considers in finding partition dimensions of graphs obtained from a subdivision operation. In particular, we derive an upper bound of partition dimension of a subdivision of a complete graph K n with n ≥ 9. Additionally for n ∈ [2, 8], we obtain the exact values of the partition dimensions.

Information Processing Letters, 2012

A graph G = (V , E) of order n is called arbitrarily partitionable, or AP for short, if given any sequence of positive integers n 1 , . . . ,n k summing up to n, we can always partition additionally G is minimal with respect to this property, i.e. it contains no AP spanning subgraph, we call it a minimal AP-graph. It has been conjectured that such graphs are sparse, i.e., there exists an absolute constant C such that |E| Cn for each of them. We construct a family of minimal AP-graphs which prove that C 1 + 1 30 (if such C exists).