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
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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