An improved procedure for colouring graphs of bounded local density

Hurley, Eoin; Verclos, Rémi de Joannis de; Kang, Ross J.

doi:10.1137/1.9781611976465.10

Cited by 17 publications

(15 citation statements)

References 22 publications

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“…A simple greedy argument shows that s 0 ðGÞ 2D 2 , and even a small improvement over the constant 2 is nontrivial. Molloy and Reed in 1997 gave an upper bound of 2 À ð ÞD 2 for % 0:002 [26], which was subsequently improved by Bruhn and Joos [6] to 1:93D 2 , by Bonamy, Perrett and Postle [5] to 1:835D 2 , and recently by Hurley, de Joannis de Verclos and Kang [23] to 1:772D 2 ; those results apply only for sufficiently large D.…”

Section: > < >mentioning

confidence: 99%

Strong Cliques in Claw-Free Graphs

Dębski

Śleszyńska-Nowak

2021

Graphs and Combinatorics

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For a graph G, $$L(G)^2$$ L ( G ) 2 is the square of the line graph of G – that is, vertices of $$L(G)^2$$ L ( G ) 2 are edges of G and two edges $$e,f\in E(G)$$ e , f ∈ E ( G ) are adjacent in $$L(G)^2$$ L ( G ) 2 if at least one vertex of e is adjacent to a vertex of f and $$e\ne f$$ e ≠ f . The strong chromatic index of G, denoted by $$s'(G)$$ s ′ ( G ) , is the chromatic number of $$L(G)^2$$ L ( G ) 2 . A strong clique in G is a clique in $$L(G)^2$$ L ( G ) 2 . Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree $$\varDelta $$ Δ is at most $$\varDelta ^2 + \frac{1}{2}\varDelta $$ Δ 2 + 1 2 Δ . This result improves the only known result $$1.125\varDelta ^2+\varDelta $$ 1.125 Δ 2 + Δ , which is a bound for the strong chromatic index of claw-free graphs.

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Section: > < >mentioning

confidence: 99%

Strong Cliques in Claw-Free Graphs

Dębski

Śleszyńska-Nowak

2021

Graphs and Combinatorics

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“…This bound is further improved for graphs with sufficiently large maximum degree. Hurley, de Joannis de Verclos and Kang [25] showed that the strong chromatic index is at most 1:772D 2 ðGÞ, for any graph G with sufficiently large maximum degree DðGÞ.…”

Section: Introductionmentioning

confidence: 99%

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Dara

Mishra

Narayanan

et al. 2022

Graphs and Combinatorics

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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.

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“…For t = 1, this is the usual chromatic index of G; for t = 2, it is known as the strong chromatic index of G, and is associated with a more famous problem of Erdős and Nešetřil [9]; for t > 2, the parameter is referred to as the distance-t chromatic index, with the study of bounded degree graphs initiated in [13]. We note that the output of Theorem 8 may be directly used as input to a recent result [11] related to Reed's conjecture [15] to bound χ(L(G) t ). This yields the following.…”

Section: Introductionmentioning

confidence: 99%

Maximising line subgraphs of diameter at most $t$

Cambie¹,

Batenburg²,

Verclos³

et al. 2021

Preprint

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We wish to bring attention to a natural but slightly hidden problem, posed by Erdős and Nešetřil in the late 1980s, an edge version of the degree-diameter problem. Our main result is that, for any graph of maximum degree ∆ with more than 1.5∆ 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 ∈ {1, 2, 3, 4, 6}. In the case ∆ = 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∆ t for graphs of large enough maximum degree ∆, markedly improving upon earlier bounds for this parameter.

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An improved procedure for colouring graphs of bounded local density

Cited by 17 publications

References 22 publications

Strong Cliques in Claw-Free Graphs

Strong Cliques in Claw-Free Graphs

Strong Edge Coloring of Cayley Graphs and Some Product Graphs

Maximising line subgraphs of diameter at most $t$

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