10.23638/DMTCS-22-1-2
https://dmtcs.episciences.org/5404
Bensmail, Julien
Julien
Bensmail
Dross, François
François
Dross
Hocquard, Hervé
Hervé
Hocquard
Sopena, Eric
Eric
Sopena
From light edges to strong edge-colouring of 1-planar graphs
A strong edge-colouring of an undirected graph $G$ is an edge-colouring where every two edges at distance at most~$2$ receive distinct colours. The strong chromatic index of $G$ is the least number of colours in a strong edge-colouring of $G$. A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the $80$'s, asserts that every graph with maximum degree $\Delta$ should have strong chromatic index at most roughly $1.25 \Delta^2$. Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly $4\Delta$, and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of $1$-planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of $1$-planar graphs with maximum degree~$\Delta$ and strong chromatic index roughly $6\Delta$. As an upper bound, we prove that the strong chromatic index of a $1$-planar graph with maximum degree $\Delta$ is at most roughly $24\Delta$ (thus linear in $\Delta$). The proof of this result is based on the existence of light edges in $1$-planar graphs with minimum degree at least~$3$.
episciences.org
Light edges
Strong edge-colouring
Strong chromatic index
1-planar graphs
[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
2020-01-09
2020-01-09
2020-01-09
en
journal article
https://hal.archives-ouvertes.fr/hal-02112188v3
1365-8050
https://dmtcs.episciences.org/5404/pdf
VoR
application/pdf
Discrete Mathematics & Theoretical Computer Science
vol. 22 no. 1
Graph Theory
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