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)

Abstract

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
Volume18
Issue number7
DOIs
Publication statusPublished - 1994

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Multigraph
Simple Graph
Graph in graph theory
Vertex of a graph
Sufficient Conditions
Range of data

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Local and global average degree in graphs and multigraphs. / Bertram, E.; Erds, P.; Horák, P.; Širáň, J.; Tuza, Z.

In: Journal of Graph Theory, Vol. 18, No. 7, 1994, p. 647-661.

Research output: Contribution to journalArticle

Bertram, E. ; Erds, P. ; Horák, P. ; Širáň, J. ; Tuza, Z. / Local and global average degree in graphs and multigraphs. In: Journal of Graph Theory. 1994 ; Vol. 18, No. 7. pp. 647-661.
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