关于Sumner-Blitch猜想的一个注记(英文)

A Note on Sumner-Blitch's Conjecture

设G是一个图.G的最小度,连通度,控制数,独立控制数和独立数分别用δ,κ,γ,i和α表示.图G是3-γ-临界的,如果γ=3,而且G增加任一条边所得的图的控制数为2.Sumner和Blitch猜想:任意连通的 3-γ-临界图满足i=3.本文证明了如果 G是使α=κ+1≤δ的连通3-γ-临界图,那么Sumner-Blitch猜想成立.

Let δ,κ,γ, i and α be the minimum degree, the connectivity, the domination number, the independent domination number and the independence number of a graph G, respectively. The graph G is 3-γ-critical if γ= 3 and the addition of any edge decreases γ by 1. It was conjectured that any connected 3-γ-critical graph satisfies i = 3. We show here that if G is a connected 3-γ-critical graph with α = κ + 1≤δ, then i = 3.