The balanced decomposition number of TK4 and series -parallel graphs

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) ≤ ⌊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.