Strong cliques and forbidden cycles

Batenburg, Wouter Cames van; Kang, Ross J.; Pirot, François

doi:10.1016/j.indag.2019.09.003

Cited by 6 publications

(11 citation statements)

References 11 publications

(25 reference statements)

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C_{5}

‐free graph

G

has strong clique index at most

normalΔ {MathClass-open(G MathClass-close)}^{2}

. This enhancement verifies the strong clique version of Conjecture 3, as well as Conjecture 4 with a much better upper bound.…”

Section: Introductionmentioning

confidence: 75%

“…Cames van Batenburg, Kang, and Pirot [6] also considered the class of graphs with other forbidden odd cycles. They proved that a

C_{3}

‐free graph

G

has strong clique index at most

1.25 normalΔ {MathClass-open(G MathClass-close)}^{2}

, which is tight for blowups of

C_{5}

.…”

Section: Introductionmentioning

confidence: 99%

“…The authors of [6] speculated that the situation is much different for the class of graphs with a forbidden even cycle. In contrast to the quadratic upper bounds of all aforementioned conjectures, they put forth the following conjecture asserting a linear upper bound:…”

Section: Introductionmentioning

confidence: 99%

“…Note that (i) in the above theorem resolves Conjecture 5 in the affirmative when

k = 2

and

normalΔ (G)

is not too small. The authors of [6] also put forth the following conjecture for bipartite graphs with a forbidden even cycle.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The strong clique index of a graph with forbidden cycles

Cho

Choi

Kim

et al. 2021

Journal of Graph Theory

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Given a graph G, the strong clique index of G, denoted ω S ( G ), is the maximum size of a set S of edges such that every pair of edges in S has distance at most 2 in the line graph of G. As a relaxation of the renowned Erdős–Nešetřil conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique index, and conjectured a quadratic upper bound in terms of the maximum degree. Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles. Namely, if G is a C 2 k‐free graph with normalΔ ( G ) ≥ max { 4 , 2 k − 2 }, then ω S ( G ) ≤ ( 2 k − 1 ) normalΔ ( G ) − )(2 k − 1 2, and if G is a C 2 k‐free bipartite graph, then ω S ( G ) ≤ k normalΔ ( G ) − ( k − 1 ). We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a { C 5 , C 2 k }‐free graph G with normalΔ ( G ) ≥ 1 satisfies ω S ( G ) ≤ k normalΔ ( G ) − ( k − 1 ), when either k ≥ 4 or k ∈ { 2 , 3 } and G is also C 3‐free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for k ≥ 3, we prove that a C 2 k‐free graph G with normalΔ ( G ) ≥ 1 satisfies ω S ( G ) ≤ ( 2 k − 1 ) normalΔ ( G ) + MathClass-open( 2 k − 1 MathClass-close) 2. This improves some results of Cames van Batenburg, Kang, and Pirot.

show abstract

C_{5}

‐free graph

G

has strong clique index at most

normalΔ {MathClass-open(G MathClass-close)}^{2}

. This enhancement verifies the strong clique version of Conjecture 3, as well as Conjecture 4 with a much better upper bound.…”

Section: Introductionmentioning

confidence: 75%

“…Cames van Batenburg, Kang, and Pirot [6] also considered the class of graphs with other forbidden odd cycles. They proved that a

C_{3}

‐free graph

G

has strong clique index at most

1.25 normalΔ {MathClass-open(G MathClass-close)}^{2}

, which is tight for blowups of

C_{5}

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Note that (i) in the above theorem resolves Conjecture 5 in the affirmative when

k = 2

and

normalΔ (G)

is not too small. The authors of [6] also put forth the following conjecture for bipartite graphs with a forbidden even cycle.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The strong clique index of a graph with forbidden cycles

Cho

Choi

Kim

et al. 2021

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…Note that this is tight as equality holds for the same graph demonstrating the tightness of Conjecture 2. It was recently revealed that forbidding all odd cycles is not necessary, as Cames van Batenburg, Kang, and Pirot [5] showed that a C 5 -free graph G has strong clique number at most ∆(G) 2 . This enhancement verifies the strong clique version of Conjecture 3, as well as Conjecture 4 with a much better upper bound.…”

Section: Introductionmentioning

confidence: 99%

The strong clique number of graphs with forbidden cycles

Cho¹,

Choi

Kim³

et al. 2020

Preprint

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Given a graph G, the strong clique number of G, denoted ω S (G), is the maximum size of a set S of edges such that every pair of edges in S has distance at most 2 in the line graph of G. As a relaxation of the renowned Erdős-Nešetřil conjecture regarding the strong chromatic index, Faudree et al. suggested investigating the strong clique number, and conjectured a quadratic upper bound in terms of the maximum degree.Recently, Cames van Batenburg, Kang, and Pirot conjectured a linear upper bound in terms of the maximum degree for graphs without even cycles., and if G is a C 2k -free bipartite graph, then ω S (G) ≤ k∆(G) − (k − 1). We prove the second conjecture in a stronger form, by showing that forbidding all odd cycles is not necessary. To be precise, we show that a {Cand G is also C 3 -free. Regarding the first conjecture, we prove an upper bound that is off by the constant term. Namely, for k ≥ 3, we prove that a C 2k -free graph G with ∆(G) ≥ 1 satisfies ω S (G) ≤ (2k − 1)∆(G) + (2k − 1) 2 . This improves some results of Cames van Batenburg, Kang, and Pirot.

show abstract