Maximising line subgraphs of diameter at most t release_h4ak7g2j5vgtnccpmfvmklvk7q

by Stijn Cambie, Wouter Cames van Batenburg, Rémi de Joannis de Verclos, Ross J. Kang

Released as a article .

2021  

Abstract

We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree–diameter problem. Our main result is that, for any graph of maximum degree Δ with more than 1.5 Δ^t edges, its line graph must have diameter larger than t. In the case where the graph contains no cycle of length 2t+1, we can improve the bound on the number of edges to one that is exact for t∈{1,2,3,4,6}. In the case Δ=3 and t=3, we obtain an exact bound. Our results also have implications for the related problem of bounding the distance-t chromatic index, t>2; in particular, for this we obtain an upper bound of 1.941Δ^t for graphs of large enough maximum degree Δ, markedly improving upon earlier bounds for this parameter.
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Type  article
Stage   submitted
Date   2021-03-22
Version   v1
Language   en ?
arXiv  2103.11898v1
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