Hamiltonian cycles in 2‐tough 2K2 $2{K}_{2}$‐free graphs

Ota, Katsuhiro; Masahiro, Sanka,

doi:10.1002/jgt.22852

Cited by 8 publications

(5 citation statements)

References 15 publications

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“…Kratsch, Lehel, and Müller showed the following theorem. It has been shown that Conjecture 1.1 is true for some superclass of split graphs, for example, spider graphs [11], chordal graphs [6,10], K 2 2 -free graphs [5,14,15], and…”

Section: Conjecture 11 (Chvátal [7] 1973mentioning

confidence: 99%

“…It has been shown that Conjecture 1.1 is true for some superclass of split graphs, for example, spider graphs [11], chordal graphs [6, 10],

2{K}_{2}

‐free graphs [5, 14, 15], and

({P}_{2}\cup {P}_{3})

‐free graphs [16]. However, some of the above results are not known to be the best about the toughness condition, which cannot be smaller than

\frac{3}{2}

by Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

Masahiro

2022

Journal of Graph Theory

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A graph G is H -free if it has no induced subgraph isomorphic to H , where H is a graph. In this paper, we show that every 3 2 -tough ∪ P P ( ) 4 1 0 -free graph has a 2-factor. The toughness condition of this result is sharp. Moreover, for any ε > 0 there exists a ε (2 − )tough P 2 5 -free graph without a 2-factor. This implies that the graph ∪ P P 4 1 0 is best possible for a forbidden subgraph in a sense. K E Y W O R D S 2-factor, Hamiltonian cycle, (P 4 ∪ P 10 )-free graph, toughness Y to denote ∩ Y N X ( ) G . If ≥ e x Y ( , ) 1 G ≥ e X Y ( ( , ) 1) G , we say that x and Y (X and Y ) are adjacent.

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Section: Conjecture 11 (Chvátal [7] 1973mentioning

confidence: 99%

“…It has been shown that Conjecture 1.1 is true for some superclass of split graphs, for example, spider graphs [11], chordal graphs [6, 10],

2{K}_{2}

‐free graphs [5, 14, 15], and

({P}_{2}\cup {P}_{3})

‐free graphs [16]. However, some of the above results are not known to be the best about the toughness condition, which cannot be smaller than

\frac{3}{2}

by Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

Masahiro

2022

Journal of Graph Theory

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“…Bauer et al [1] proved that every 3 2 -tough 5-chordal graph with at least three vertices has a 2-factor [1], where a 5-chordal graph is one with no induced cycles of length larger than 5. This implies that every 3 2 -tough 2P 2free graph on at least three vertices has a 2-factor, see [10]. Sanka [12] proved that every 3 2 -tough (P 10 ∪ P 4 )-free graph has a 2-factor.…”

Section: Introductionmentioning

confidence: 96%

Comparison of Contents Design in Mathematical Inquiry in High School Textbooks

2021

School Mathematics Textbooks in China

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Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order n with diameter d ≥ 2 that is not a path, the number of Laplacian eigenvalues in the interval [n − d + 2, n] is at most n − d. We show that the conjecture is true and provide a family of graphs to show that the bound is best possible. This establishes an interesting relation between the spectral and classical parameters.

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“…Chvátal's toughness conjecture has been verified for certain classes of graphs including planar graphs, claw-free graphs, co-comparability graphs, and chordal graphs [2]. The classes also include 2K 2 -free graphs [6,15,13], and R-free graphs for R ∈ {P 2 ∪ P 3 , P 3 ∪ 2P 1 , P 2 ∪ kP 1 } [16,9,17,12,19], where k 4 is an integer. In general, the conjecture is still wide open.…”

Section: Introductionmentioning

confidence: 99%

An Ore-Type Condition for Hamiltonicity in Tough Graphs and the Extremal Examples

Sanka,

Shan

2024

Electron. J. Combin.

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Let $G$ be a $t$-tough graph on $n\ge 3$ vertices for some $t>0$. It was shown by Bauer et al. in 1995 that if the minimum degree of $G$ is greater than $\frac{n}{t+1}-1$, then $G$ is hamiltonian. In terms of Ore-type hamiltonicity conditions, the problem was only studied when $t$ is between 1 and 2, and recently the second author proved a general result. The result states that if the degree sum of any two nonadjacent vertices of $G$ is greater than $\frac{2n}{t+1}+t-2$, then $G$ is hamiltonian. It was conjectured in the same paper that the "$+t$" in the bound $\frac{2n}{t+1}+t-2$ can be removed. Here we confirm the conjecture. The result generalizes the result by Bauer, Broersma, van den Heuvel, and Veldman. Furthermore, we characterize all $t$-tough graphs $G$ on $n\ge 3$ vertices for which $\sigma_2(G) = \frac{2n}{t+1}-2$ but $G$ is non-hamiltonian.

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Hamiltonian cycles in 2‐tough 2K2 $2{K}_{2}$‐free graphs

Cited by 8 publications

References 15 publications

Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

Forbidden subgraphs and 2‐factors in 3/2‐tough graphs

Comparison of Contents Design in Mathematical Inquiry in High School Textbooks

An Ore-Type Condition for Hamiltonicity in Tough Graphs and the Extremal Examples

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