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On graphs G for which both g and G̅ are claw-free

Shinya Fujita — 2005

Discussiones Mathematicae Graph Theory

Let G be a graph with |V(G)| ≥ 10. We prove that if both G and G̅ are claw-free, then minΔ(G), Δ(G̅) ≤ 2. As a generalization of this result in the case where |V(G)| is sufficiently large, we also prove that if both G and G̅ are K 1 , t -free, then minΔ(G),Δ(G̅) ≤ r(t- 1,t)-1 where r(t-1,t) is the Ramsey number.

The Balanced Decomposition Number of TK4 and Series-Parallel Graphs

Shinya FujitaHenry Liu — 2013

Discussiones Mathematicae Graph Theory

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)...

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