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Journal Articles Journal of Graph Theory Year : 2019

Upper bounds on the uniquely restricted chromatic index

Julien Baste
Ignasi Sau

Abstract

Golumbic, Hirst, and Lewenstein define a matching in a simple, finite, and undirected graph G to be uniquely restricted if no other matching covers exactly the same set of vertices. We consider uniquely restricted edge-colorings of G defined as partitions of its edge set into uniquely restricted matchings, and study the uniquely restricted chromatic index χ′ur(G) of G, defined as the minimum number of uniquely restricted matchings required for such a partition. For every graph G, χ′(G) ≤ a′(G) ≤ χ′ur(G) ≤ χ′s(G), where χ′(G) is the classical chromatic index, a′(G) is the acyclic chromatic index, and χ′s(G) is the strong chromatic index of G, respectively. While Vizing’s famous theorem states that χ′(G) is either the maximum degree ∆(G) of G or ∆(G) + 1, two famous open conjectures due to Alon, Sudakov, and Zaks, and to Erdo ̋s and Neˇsetˇril concern upper bounds on a′(G) and χ′s(G) in terms of ∆(G). Since χ′ur(G) is sandwiched between these two parameters, studying upper bounds in terms of ∆(G) is a natural problem. We show that χ′ur(G) ≤ ∆(G)2 with equality if and only if some component of G is K∆(G),∆(G). If G is connected, bipartite, and distinct from K∆(G),∆(G), and ∆(G) is at least 4, then, adapting Lova ́sz’s elegant proof of Brooks’ theorem, we show that χ′ur (G) ≤ ∆(G)2 − ∆(G). Our proofs are constructive and yield efficient algorithms to determine the corresponding edge-colorings.
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Dates and versions

lirmm-02412578 , version 1 (15-12-2019)

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Julien Baste, Dieter Rautenbach, Ignasi Sau. Upper bounds on the uniquely restricted chromatic index. Journal of Graph Theory, 2019, 91 (3), pp.251-258. ⟨10.1002/jgt.22429⟩. ⟨lirmm-02412578⟩
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