In this paper, we prove that if any set of ｜E（G）｜- ｜V（G）｜ ＋ 1 facial cycles of a 3-connected planar graph G embedded in the plane doesn＇t form a minimum cycle base of G, then any minimum cycle base of G cont...In this paper, we prove that if any set of ｜E（G）｜- ｜V（G）｜ ＋ 1 facial cycles of a 3-connected planar graph G embedded in the plane doesn＇t form a minimum cycle base of G, then any minimum cycle base of G contains a separating cycle, and G has a minor isomorphic to T6, where T6 is the graph obtained from the complete graph K6 by deleting a path with four edges.展开更多
Let S = （a1,...,am;b1,...,bn）, where a1,...,am and b1,...,bn are two nonincreasing sequences of nonnegative integers. The pair S= （a1,..., am; b1,..., bn） is said to be a bigraphic pair if there is a simple bipart...Let S = （a1,...,am;b1,...,bn）, where a1,...,am and b1,...,bn are two nonincreasing sequences of nonnegative integers. The pair S= （a1,..., am; b1,..., bn） is said to be a bigraphic pair if there is a simple bipartite graph G = （X ∪ Y, E） such that a1,…, am and b1,..., bn are the degrees of the vertices in X and Y, respectively. Let Z3 be the cyclic group of order 3. Define σ（Z3, m, n） to be the minimum integer k such that every bigraphic pair S = （a1,..., am; b1,..., bn） with am, b ≥ 2 and σ（S） = a1 ＋ ... ＋ am ≥ k has a Z3-connected realization. For n = m, Yin [Discrete Math., 339, 2018-2026 （2016）] recently determined the values of σ（Z3,m,m） for m ≥ 4. In this paper, we completely determine the values of σ（Z3, m, n） for m ≥ n ≥4.展开更多
文摘In this paper, we prove that if any set of ｜E（G）｜- ｜V（G）｜ ＋ 1 facial cycles of a 3-connected planar graph G embedded in the plane doesn＇t form a minimum cycle base of G, then any minimum cycle base of G contains a separating cycle, and G has a minor isomorphic to T6, where T6 is the graph obtained from the complete graph K6 by deleting a path with four edges.
基金Supported by National Natural Science Foundation of China(Grant No.11561017)Natural Science Foundation of Hainan Province(Grant No.2016CXTD004)
文摘Let S = （a1,...,am;b1,...,bn）, where a1,...,am and b1,...,bn are two nonincreasing sequences of nonnegative integers. The pair S= （a1,..., am; b1,..., bn） is said to be a bigraphic pair if there is a simple bipartite graph G = （X ∪ Y, E） such that a1,…, am and b1,..., bn are the degrees of the vertices in X and Y, respectively. Let Z3 be the cyclic group of order 3. Define σ（Z3, m, n） to be the minimum integer k such that every bigraphic pair S = （a1,..., am; b1,..., bn） with am, b ≥ 2 and σ（S） = a1 ＋ ... ＋ am ≥ k has a Z3-connected realization. For n = m, Yin [Discrete Math., 339, 2018-2026 （2016）] recently determined the values of σ（Z3,m,m） for m ≥ 4. In this paper, we completely determine the values of σ（Z3, m, n） for m ≥ n ≥4.
