Maximising line subgraphs of diameter at most $t$

TL;DR

This paper investigates the maximum number of edges in graphs with bounded degree and diameter, establishing bounds that influence the understanding of line graph diameters and related coloring problems.

Contribution

It provides new bounds on the number of edges in graphs with given degree and diameter constraints, including exact bounds for specific cases and implications for the distance-$t$ chromatic index.

Findings

Graphs with more than 1.5Δ^t edges have line graphs with diameter exceeding t.

Improved bounds for graphs without cycles of length 2t+1, exact for t=1,2,3,4,6.

Upper bound of 1.941Δ^t for the distance-$t$ chromatic index in large degree graphs.

Abstract

We wish to bring attention to a natural but slightly hidden problem, posed by Erd\H{o}s and Ne\v{s}et\v{r}il in the late 1980s, an edge version of the degree--diameter problem. Our main result is that, for any graph of maximum degree $\Delta$ with more than $1.5 \Delta^t$ 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 $t\in\{1,2,3,4,6\}$. In the case $\Delta=3$ 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 $1.941\Delta^t$ for graphs of large enough maximum degree $\Delta$, markedly improving upon earlier bounds for this parameter.