Local and global average degree in graphs and multigraphs

E. Bertram, P. Erds, P. Horák, J. Širáň, Z. Tuza

Research output: Contribution to journalArticle

6 Citations (Scopus)


A vertex v of a graph G is called groupie if the average degree tv of all neighbors of v in G is not smaller than the average degree tG of G. Every graph contains a groupie vertex; the problem of whether or not every simple graph on ≧2 vertices has at least two groupie vertices turned out to be surprisingly difficult. We present various sufficient conditions for a simple graph to contain at least two groupie vertices. Further, we investigate the function f(n) = max minv (tv/tG), where the maximum ranges over all simple graphs on n vertices, and prove that f(n) = 1/42n + o(1). The corresponding result for multigraphs is in sharp contrast with the above. We also characterize trees in which the local average degree tv is constant.

Original languageEnglish
Pages (from-to)647-661
Number of pages15
JournalJournal of Graph Theory
Issue number7
Publication statusPublished - 1994


ASJC Scopus subject areas

  • Geometry and Topology

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