Abstract
We present upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity with respect to order and size. All bounds presented in this paper are asymptotically sharp.
Highlights
Our motivation for this paper comes from the results published by Erdős et al in [4] and Mukwembi in [5].Graphs with forbidden subgraphs are a big part of graph theory literature such as in [4,5,6,7,8]
We are concerned, in part, with upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity in terms of order. e diameter is the most common of the classical distance parameters in graph theory, and much of the research on distances is on the diameter [9]
We provided tight upper bounds for bipartite graphs with respect to order and with respect to size for any value of λ
Summary
Our motivation for this paper comes from the results published by Erdős et al in [4] and Mukwembi in [5]. For positive integers a and b, if ab ≥ 5, a + b ≥ 5 It is the purpose of this paper to bound the diameter of any triangle-free graph with respect to order and edge connectivity. Let G be a λ-edge-connected graph, λ ∈ {3, 4}, of order n; diam(G) ≤ ⌊(n − 1)/2⌋ This inequality is best possible with the exception of a small constant. Let G be a λ-edge-connected, λ ≥ 6, triangle-free graph of size m; diam(G) ≤ (4m/λ2) + 1 Let G be a λ-edge-connected bipartite graph of size m; diam(G) ≤ (4m/λ2) + 1 The cases where λ 3 and λ 5 need more care and as such have been allotted space among our main results
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