Maximal triangle‐free graphs with restrictions on the degrees

Z. Füredi, Ákos Seress

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)

Original languageEnglish
Pages (from-to)11-24
Number of pages14
JournalJournal of Graph Theory
Volume18
Issue number1
DOIs
Publication statusPublished - 1994

Fingerprint

Restriction
Graph in graph theory
Exception
Upper and Lower Bounds
Denote
Interval

ASJC Scopus subject areas

  • Geometry and Topology

Cite this

Maximal triangle‐free graphs with restrictions on the degrees. / Füredi, Z.; Seress, Ákos.

In: Journal of Graph Theory, Vol. 18, No. 1, 1994, p. 11-24.

Research output: Contribution to journalArticle

@article{994c731da51040e3bb224ddd3141cd34,
title = "Maximal triangle‐free graphs with restrictions on the degrees",
abstract = "We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)",
author = "Z. F{\"u}redi and {\'A}kos Seress",
year = "1994",
doi = "10.1002/jgt.3190180103",
language = "English",
volume = "18",
pages = "11--24",
journal = "Journal of Graph Theory",
issn = "0364-9024",
publisher = "Wiley-Liss Inc.",
number = "1",

}

TY - JOUR

T1 - Maximal triangle‐free graphs with restrictions on the degrees

AU - Füredi, Z.

AU - Seress, Ákos

PY - 1994

Y1 - 1994

N2 - We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)

AB - We investigate the problem that at least how many edges must a maximal triangle‐free graph on n vertices have if the maximal valency is ≤D. Denote this minimum value by F(n, D). For large enough n, we determine the exact value of F(n, D) if D ≥ (n − 2)/2 and we prove that lim F(n, cn)/n = K(c) exists for all 0 < c with the possible exception of a sequence ck → 0. The determination of K(c) is a finite problem on all intervals [γ, ∞). For D = cnϵ, 1/2 < ϵ < 1, we give upper and lower bounds for F(n, D) differing only in a constant factor. (Clearly, D < (n ‐ 1)1/2 is impossible in a maximal triangle‐free graph.)

UR - http://www.scopus.com/inward/record.url?scp=0042190006&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0042190006&partnerID=8YFLogxK

U2 - 10.1002/jgt.3190180103

DO - 10.1002/jgt.3190180103

M3 - Article

AN - SCOPUS:0042190006

VL - 18

SP - 11

EP - 24

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 1

ER -