On size, order, diameter and edge-connectivity of graphs

Ali, Patrick; Mazorodze, Jaya Percival; Mukwembi, Simon; Vetrík, Tomáš

doi:10.1007/s10474-017-0699-1

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Hence, the vertex c 2 has exactly two neighbors in V i . First, suppose that Next, we assume that (1,2), or (2, 1), then some vertex in V i+1 has two neighbors in V i that have a common neighbor in V i−1 , which is a contradiction. This implies (7) in this case.…”

Section: Proofsmentioning

confidence: 99%

“…that is, asymptotically, there is only little room for improvements of (1). If λ ≥ 3, then δ ≥ 3, and (1) implies d ≤ 5n 8 , which was recently improved by Fundikwa et al [11] to…”

mentioning

confidence: 98%

“…+ O(1) has been discovered several times [2,9,14,15]. Lower bounds on the order of graphs realizing a certain diameter subject to conditions such as triangle-freeness [9,12], C 4 -freeness [9,11], or conditions involving the chromatic number [5,6] or the edge-connectivity have been studied [1,11,12]. Similar problems for notions related to the diameter also received attention [7,8,13].…”

mentioning

confidence: 99%

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Diameter, edge-connectivity, and $C_4$-freeness

Vanessa¹,

Pardey²,

Rautenbach³

2021

Preprint

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Improving a recent result of Fundikwa, Mazorodze, and Mukwembi, we show that d ≤ (2n−3)/5 for every connected C 4 -free graph of order n, diameter d, and edge-connectivity at least 3, which is best possible up to a small additive constant. For edge-connectivity at least 4, we improve this to d ≤ (n − 3)/3. Furthermore, adapting a construction due to Erdős, Pach, Pollack, and Tuza, for an odd prime power q at least 7, and every positive integer k, we show the existence of a connected C 4 -free graph of order n = (q 2 + q − 1)k + 1, diameter d = 4k, and edge-connectivity λ at least q − 6, in particular, d ≥ 4(n − 1)/(λ 2 + O(λ)).

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Section: Proofsmentioning

confidence: 99%

“…that is, asymptotically, there is only little room for improvements of (1). If λ ≥ 3, then δ ≥ 3, and (1) implies d ≤ 5n 8 , which was recently improved by Fundikwa et al [11] to…”

mentioning

confidence: 98%