On the clique number of the square of a line graph and its relation to maximum degree of the line graph

Faron, Maxime; Postle, Luke

doi:10.1002/jgt.22452

Cited by 10 publications

(8 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A common element of the proof is a lemma about the Ore-degree σ G of G, the largest over all edges of G of the sum of the two endpoint degrees. The following generalises a recent result due to Faron and Postle [4].…”

Section: No Triangles or No 5-cyclessupporting

confidence: 85%

Strong cliques and forbidden cycles

Batenburg

Kang

Pirot

2020

Indagationes Mathematicae

View full text Add to dashboard Cite

Given a graph G, the strong clique number ω 2 (G) of G is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in G. We study the strong clique number of graphs missing some set of cycle lengths. For a graph G of large enough maximum degree ∆, we show among other results the following:These bounds are attained by natural extremal examples. Our work extends and improves upon previous work of Faudree, Gyárfás, Schelp and Tuza (1990), Mahdian (2000) and Faron and Postle (2019). We are motivated by the corresponding problems for the strong chromatic index.

show abstract

Section: No Triangles or No 5-cyclessupporting

confidence: 85%

Strong cliques and forbidden cycles

Batenburg

Kang

Pirot

2020

Indagationes Mathematicae

View full text Add to dashboard Cite

show abstract

“…We should remark that Dȩbski and Śleszyńska-Nowak [17] announced a bound of roughly 7 4 ∆ t . Note that the bound in Theorem 8 can be improved in the cases t ∈ {1, 2}: ω(L(G)) ≤ ∆ + 1 is trivially true, while ω(L(G) 2 ) ≤ 4 3 ∆ 2 is a recent result of Faron and Postle [10]. We also have a bound on ω(L(G) t ) analogous to Theorem 7, a result stated and shown in Section 2.…”

Section: Introductionmentioning

confidence: 64%

Maximising line subgraphs of diameter at most $t$

Cambie¹,

Batenburg²,

Verclos³

et al. 2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…Conjecture 2 remains open, but is closer to being proved than Conjecture 1. The second author proved that x L 2 ðGÞ ð Þ 3 2 D 2 [28], which was improved to 4 3 D 2 by Faron and Postle [14]. There are also sharp results for specific classes of graphs: for bipartite graphs the upper bound is D 2 [16] (sharp for complete bipartite graphs), and for triangle-free graphs it is 5 4 D 2 [3, Theorem 4.1.5] (sharp for blowups of C 5 ).…”

Section: > < >mentioning

confidence: 96%

Strong Cliques in Claw-Free Graphs

Dębski

Śleszyńska-Nowak

2021

Graphs and Combinatorics

View full text Add to dashboard Cite

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.

show abstract

On the clique number of the square of a line graph and its relation to maximum degree of the line graph

Cited by 10 publications

References 6 publications

Strong cliques and forbidden cycles

Strong cliques and forbidden cycles

Maximising line subgraphs of diameter at most $t$

Strong Cliques in Claw-Free Graphs

Contact Info

Product

Resources

About