摘要
A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n+3-i/3where 1 ≤i≤6, i=n (mod 6) andn≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, f2(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.
A 2-graph is a hypergraph with edge sizes of at most two. A regular 2-graph is said to be minimal if it does not contain a proper regular factor. Let f2(n) be the maximum value of degrees over all minimal regular 2-graphs of n vertices. In this paper, we provide a structure property of minimal regular 2-graphs, and consequently, prove that f2(n) = n+3-i/3, where 1 ≤ i ≤ 6, i ≡ n (mod 6) and n ≥ 7, which solves a conjecture posed by Fan, Liu, Wu and Wong. As applications in graph theory, we are able to characterize unfactorable regular graphs and provide the best possible factor existence theorem on degree conditions. Moreover, fa(n) and the minimal 2-graphs can be used in the universal switch box designs, which originally motivated this study.
作者
FAN Hongbing, LIU Guizhen & LIU Jiping Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON., N2L 3C5 Canada
School of Mathematics and System Science, Shandong University, Jinan 250100, China
Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, AB., T1K 3M4, Canada
基金
supported by the Natural Sciences and Engineering Research Council of Canada
the National Natural Science Foundation of China(Grant No.10471078)
Specialied Research Fund for the Doctoral Program of Higher Education(Grant No.20040422004)of China.