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On the algorithmic complexity of determining the avd and nsd chromatic indices of graphs

Julien Bensmail 1 Hervé Hocquard 2 Dimitri Lajou 2
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
Laboratoire I3S - COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués, CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : A proper k-edge-colouring φ of a graph G is an assignment of colours from {1,...,k} to the edges of G such that no two adjacent edges receive the same colour. If, additionally, φ guarantees that no two adjacent vertices of G are incident to the same sets or sums of colours, then φ is called an avd or nsd edge-colouring, respectively. The chromatic index χ(G) of G is the smallest k such that proper k-edge-colourings of G exist. Similarly, the avd and nsd chromatic indices χavd(G) and χnsd(G) of G are the smallest k such that avd and nsd k-edge-colourings of G exist, respectively. These chromatic parameters are quite related, as we always have χnsd(G)≥χavd(G)≥χ(G). By a well-known result of Vizing, we know that, for any graph G, we must have χ(G)∈{∆(G),∆(G)+1}. Still, determining χ(G) is NP-hard in general. Regarding χavd(G) and χnsd(G), it is conjectured that, in general, they should always lie in {∆(G),∆(G)+1,∆(G)+2}. In this work, we prove that determining whether a given graph with maximum degree ∆ has avd or nsd chromatic index ∆ is NP-hard for every ∆≥3. We also prove that, for a given graph with maximum degree ∆, determining whether the avd or nsd chromatic index is ∆+1 is NP-hard for every ∆≥3. Through other NP-hardness results, we also establish that there are infinitely many graphs for which the avd and nsd chromatic indices are different. We actually come up, for every ∆≥4, with infinitely many graphs with maximum degree ∆, avd chromatic index ∆, and nsd chromatic index ∆+1, and similarly, for every ∆≥3, with infinitely many graphs with maximum degree ∆, avd chromatic index ∆+1, and nsd chromatic index ∆+2. In both cases, recognising graphs having those properties is actually NP-hard.
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https://hal.archives-ouvertes.fr/hal-03609262
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Submitted on : Tuesday, March 15, 2022 - 2:22:31 PM
Last modification on : Wednesday, March 16, 2022 - 3:38:28 AM

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Julien Bensmail, Hervé Hocquard, Dimitri Lajou. On the algorithmic complexity of determining the avd and nsd chromatic indices of graphs. [Research Report] Université Côte d'Azur; Université de Bordeaux. 2022. ⟨hal-03609262⟩

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