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Discussiones Mathematicae Graph Theory

The Balanced Decomposition Number of TK4 and Series-parallel Graphs

 Shinya Fujita Department of Integrated Design Engineering Maebashi Institute of Technology 460-1 Kamisadori, Maebashi, 371-0816, Japan Henry Liu Departamento de Matemática Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa Quinta da Torre, 2829-516 Caparica, Portugal

Abstract

A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, say red and blue, such that there is the same number of vertices in each colour. The balanced decomposition number f(G) of G is the minimum integer s with the following property: For any balanced colouring of G, there is a partition V(G) = V1 ⨃   … ⨃  Vr such that, for every i, Vi induces a connected subgraph of order at most s, and contains the same number of red and blue vertices. The function f(G) was introduced by Fujita and Nakamigawa in 2008. They conjectured that f(G) ≤ ⎣n/2 ⎦+1 if G is a 2-connected graph on n vertices. In this paper, we shall prove two partial results, in the cases when G is a subdivided K4, and a 2-connected series-parallel graph.

Keywords: graph decomposition, vertex colouring, k-connected

2010 Mathematics Subject Classification: 05C15, 05C40, 05C70.

References

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