The Second Neighbourhood for Bipartite Tournaments

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Abstract

Let T (XY, A) be a bipartite tournament with partite sets X, Y and arc set A. For any vertex xXY, the second out-neighbourhood N++(x) of x is the set of all vertices with distance 2 from x. In this paper, we prove that T contains at least two vertices x such that |N++(x)| ≥ |N+(x)| unless T is in a special class ℬ1 of bipartite tournaments; show that T contains at least a vertex x such that |N++(x)| ≥ |N(x)| and characterize the class ℬ2 of bipartite tournaments in which there exists exactly one vertex x with this property; and prove that if |X| = |Y | or |X| ≥ 4|Y |, then the bipartite tournament T contains a vertex x such that |N++(x)|+|N+(x)| ≥ 2|N(x)|.

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Discussiones Mathematicae Graph Theory

The Journal of University of Zielona Góra

Journal Information


IMPACT FACTOR 2017: 0.601
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CiteScore 2017: 0.64

SCImago Journal Rank (SJR) 2017: 0.633
Source Normalized Impact per Paper (SNIP) 2017: 1.095

Mathematical Citation Quotient (MCQ) 2017: 0.36

Target Group

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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