A Stronger Bound for the Strong Chromatic Index

Bruhn, Henning; Joos, Felix

doi:10.1017/s0963548317000244

Cited by 47 publications

(61 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Erdős and Nešetřil (see ) conjectured

χ'_{-2.3pt s} (G) \leq \frac{5}{4} {normalΔ}^{2}

, and much of the research on the strong chromatic index is motivated by this conjecture. Building on earlier work of Molloy and Reed and Bruhn and Joos , Bonamy et al showed

χ'_{-2.3pt s} (G) \leq 1.835 {normalΔ}^{2}

provided that

normalΔ

is sufficiently large. For further results on the strong chromatic index, we refer to .…”

Section: Introductionmentioning

confidence: 92%

Upper bounds on the uniquely restricted chromatic index

Baste

Rautenbach

2018

Journal of Graph Theory

View full text Add to dashboard Cite

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 ′ -2.6ptnormalu normalr ( 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 ′ normalu normalr ( G ) ≤ χ ′ normals ( G ) , where χ ′ ( G ) is the classical chromatic index, a ′ ( G ) the acyclic chromatic index, and χ ′ normals ( G ) the strong chromatic index of G. While Vizing's famous theorem states that χ ′ ( G ) is either the maximum degree normalΔ ( G ) of G or normalΔ ( G ) + 1, two famous open conjectures due to Alon, Sudakov, and Zaks, and to Erdős and Nešetřil concern upper bounds on a ′ ( G ) and χ ′ -2.14pts ( G ) in terms of normalΔ ( G ). Since χ ′ -2.4ptnormalu normalr ( G ) is sandwiched between these two parameters, studying upper bounds in terms of normalΔ ( G ) is a natural problem. We show that χ ′ -2.4ptnormalu normalr ( G ) ≤ Δ ( G ) 2 with equality if and only if some component of G is K normalΔ ( G ) , normalΔ ( G ). If G is connected, bipartite, and distinct from K normalΔ ( G ) , normalΔ ( G ), and normalΔ ( G ) is at least 4, then, adapting Lovász's elegant proof of Brooks’ theorem, we show that χ ′ -2.4ptnormalu normalr ( G ) ≤ Δ ( G ) 2 − Δ ( G ). Our proofs are constructive and yield efficient algorithms to determine the corresponding edge‐colorings.

show abstract

“…Erdős and Nešetřil (see ) conjectured

χ'_{-2.3pt s} (G) \leq \frac{5}{4} {normalΔ}^{2}

, and much of the research on the strong chromatic index is motivated by this conjecture. Building on earlier work of Molloy and Reed and Bruhn and Joos , Bonamy et al showed

χ'_{-2.3pt s} (G) \leq 1.835 {normalΔ}^{2}

provided that

normalΔ

is sufficiently large. For further results on the strong chromatic index, we refer to .…”

Section: Introductionmentioning

confidence: 92%

Upper bounds on the uniquely restricted chromatic index

Baste

Rautenbach

2018

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Next we need a concentration inequality of Bruhn and Joos [4], derived from Talagrand's inequality [20]. To that end, we introduce the following definition.…”

Section: Probabilistic Toolsmentioning

confidence: 99%

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

Kelly¹,

Kühn²,

Osthus³

2021

Preprint

View full text Add to dashboard Cite

A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if G1, . . . , Gm are nearly disjoint graphs of maximum degree at most D, then the following holds. For every fixed C, if each vertex v ∈ m i=1 V (Gi) is contained in at most C of the graphs G1, . . . , Gm, then the (list) chromatic number of m i=1 Gi is at most D + o(D). This result confirms a special case of a conjecture of Vu and generalizes Kahn's bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy, and we derive this result from a more general list coloring result in the setting of 'color degrees' that also implies a result of Reed and Sudakov.

show abstract

“…Bruhn and Joos [9] proved that v 0 s ðGÞ 1:93D 2 ðGÞ, for graphs of sufficiently large maximum degree. This improves an old bound of v 0 s ðGÞ 1:998D 2 ðGÞ proved by Molloy and Reed [32].…”

Section: Introductionmentioning

confidence: 99%

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Dara

Mishra

Narayanan

et al. 2022

Graphs and Combinatorics

View full text Add to dashboard Cite

A strong edge coloring of a graph G is a proper edge coloring of G such that every color class is an induced matching. The minimum number of colors required is termed the strong chromatic index. In this paper we determine the exact value of the strong chromatic index of all unitary Cayley graphs. Our investigations reveal an underlying product structure from which the unitary Cayley graphs emerge. We then go on to give tight bounds for the strong chromatic index of the Cartesian product of two trees, including an exact formula for the product in the case of stars. Further, we give bounds for the strong chromatic index of the product of a tree with a cycle. For any tree, those bounds may differ from the actual value only by not more than a small additive constant (at most 2 for even cycles and at most 4 for odd cycles), moreover they yield the exact value when the length of the cycle is divisible by 4.

show abstract

A Stronger Bound for the Strong Chromatic Index

Cited by 47 publications

References 26 publications

Upper bounds on the uniquely restricted chromatic index

Upper bounds on the uniquely restricted chromatic index

A special case of Vu's conjecture: Coloring nearly disjoint graphs of bounded maximum degree

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Contact Info

Product

Resources

About