### Abstract

Assume that G is a triangle-free graph. Let h_{d}(G) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that h_{2}(G) = θ(n log n) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of h_{2}(G) and the clique-cover number of edges of graphs. It is also shown that the maximum value of h_{2}(G) over (triangle-free) graphs of order n is ⌈n/2 - 1⌉⌊n/2 - 1⌋. The behavior of h_{3}(G) is different, its maximum value is n - 1. We could not decide whether h_{4}(G)≤(1 - ∈)n for connected (triangle-free) graphs of order n with a positive ∈.

Original language | English |
---|---|

Pages (from-to) | 493-501 |

Number of pages | 9 |

Journal | Combinatorica |

Volume | 18 |

Issue number | 4 |

Publication status | Published - 1998 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Discrete Mathematics and Combinatorics

### Cite this

*Combinatorica*,

*18*(4), 493-501.

**How to decrease the diameter of triangle-free graphs.** / Erdős, P.; Gyárfás, A.; Ruszinkó, Miklós.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 18, no. 4, pp. 493-501.

}

TY - JOUR

T1 - How to decrease the diameter of triangle-free graphs

AU - Erdős, P.

AU - Gyárfás, A.

AU - Ruszinkó, Miklós

PY - 1998

Y1 - 1998

N2 - Assume that G is a triangle-free graph. Let hd(G) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that h2(G) = θ(n log n) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of h2(G) and the clique-cover number of edges of graphs. It is also shown that the maximum value of h2(G) over (triangle-free) graphs of order n is ⌈n/2 - 1⌉⌊n/2 - 1⌋. The behavior of h3(G) is different, its maximum value is n - 1. We could not decide whether h4(G)≤(1 - ∈)n for connected (triangle-free) graphs of order n with a positive ∈.

AB - Assume that G is a triangle-free graph. Let hd(G) be the minimum number of edges one has to add to G to get a graph of diameter at most d which is still triangle-free. It is shown that h2(G) = θ(n log n) for connected graphs of order n and of fixed maximum degree. The proof is based on relations of h2(G) and the clique-cover number of edges of graphs. It is also shown that the maximum value of h2(G) over (triangle-free) graphs of order n is ⌈n/2 - 1⌉⌊n/2 - 1⌋. The behavior of h3(G) is different, its maximum value is n - 1. We could not decide whether h4(G)≤(1 - ∈)n for connected (triangle-free) graphs of order n with a positive ∈.

UR - http://www.scopus.com/inward/record.url?scp=0040689686&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040689686&partnerID=8YFLogxK

M3 - Article

VL - 18

SP - 493

EP - 501

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -