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Rainbow colouring of split and threshold Graphs

Chandran, Sunil L and Rajendraprasad, Deepak (2012) Rainbow colouring of split and threshold Graphs. In: Computing and Combinatorics, 7434 . pp. 181-192.

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Official URL: http://dx.doi.org/10.1007/978-3-642-32241-9_16

Abstract

A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chordal Graph is a graph in which every cycle of length more than 3 has a chord. A Split Graph is a chordal graph whose vertices can be partitioned into a clique and an independent set. A threshold graph is a split graph in which the neighbourhoods of the independent set vertices form a linear order under set inclusion. In this article, we show the following: 1. The problem of deciding whether a graph can be rainbow coloured using 3 colours remains NP-complete even when restricted to the class of split graphs. However, any split graph can be rainbow coloured in linear time using at most one more colour than the optimum. 2. For every integer k ≥ 3, the problem of deciding whether a graph can be rainbow coloured using k colours remains NP-complete even when restricted to the class of chordal graphs. 3. For every positive integer k, threshold graphs with rainbow connection number k can be characterised based on their degree sequence alone. Further, we can optimally rainbow colour a threshold graph in linear time.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Springer Berlin Heidelberg.
Keywords: Rainbow Connectivity; Rainbow Colouring; Threshold Graphs; Split Graphs; Chordal Graphs; Degree Sequence; Approximation; Complexity
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Depositing User: Francis Jayakanth
Date Deposited: 04 Nov 2013 09:11
Last Modified: 04 Nov 2013 09:11
URI: http://eprints.iisc.ac.in/id/eprint/47703

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