The Balanced Decomposition Number of TK4 and Series-Parallel Graphs

• Autorzy: Shinya Fujita , Henry Liu
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

graph decomposition; vertex colouring; k-connected

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2013
• Tom: 33
• Numer: 2
• Strony: 347-359

Daty

wydano

2013-05-01

online

2013-04-13

Twórcy

autor

Shinya Fujita

• Department of Integrated Design Engineering Maebashi Institute of Technology 460-1 Kamisadori, Maebashi, 371-0816, Japan

autor

Henry Liu

• Departamento de Matemática Faculdade de Ciˆencias e Tecnologia Universidade Nova de Lisboa Quinta da Torre, 2829-516 Caparica, Portugal

Bibliografia

• [1] B. Bollobás, Modern Graph Theory (Springer-Verlag, New York, 1998).
• [2] R.J. Duffin, Topology of series-parallel networks, J. Math. Anal. Appl. 10 (1965) 303-318.
• [3] E.S. Elmallah and C.J. Colbourn, Series-parallel subgraphs of planar graphs, Networks 22 (1992) 607-614. doi:10.1002/net.3230220608[Crossref]
• [4] S. Fujita and H. Liu, The balanced decomposition number and vertex connectivity, SIAM. J. Discrete Math. 24 (2010) 1597-1616. doi:10.1137/090780894[WoS][Crossref]
• [5] S. Fujita and H. Liu, Further results on the balanced decomposition number , in: Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing, Congr. Numer. 202 (2010) 119-128.
• [6] S. Fujita and T. Nakamigawa, Balanced decomposition of a vertex-coloured graph, Discrete Appl. Math. 156 (2008) 3339-3344. doi:10.1016/j.dam.2008.01.006[Crossref]

Identyfikatory

DOI

10.7151/dmgt.1666