Maximising line subgraphs of diameter at most $t$

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 ∆ with more than 1.5∆ 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 ∆ = 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 1.941∆ t for graphs of large enough maximum degree ∆, markedly improving upon earlier bounds for this parameter.

Coloring, List Coloring, and Painting Squares of Graphs (and Other Related Problems)

“…Chung, Gyárfás, Tuza, and Trotter exactly determined h 2 (∆); it is 5 4 ∆ 2 +1 when ∆ is even and slightly smaller when ∆ is odd (see the start of Section 4.1). For larger k, Cambie, Cames van Batenburg, de Joannis de Verclos, and Kang [44] showed that ω k (L(G)) ⩽ 3 2 ∆ k . In particular, h k (∆) ⩽ 3 2 ∆ k + 1.…”