Remarks on path factors in graphs

Zhou, Sizhong

doi:10.1051/ro/2019111

Cited by 35 publications

(21 citation statements)

References 24 publications

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“…A graph

G κ (G) \geq 2 m + 1

and

b i n d (G) > \frac{3 m + 1}{2 m + 1}

is a

(P_{\geq 3}, m)

‐factor deleted graph, where

m \in double-struckN

. The bound of binding number improves the given theorem in Zhou 17 …”

Section: Conclusion and Discussionsupporting

confidence: 51%

“… Zhou et al 22 proved that a graph

G

with

κ (G) \geq k + 2

is a

(P_{\geq 3}, k)

‐factor critical graph if

t (G) \geq \frac{k + 1}{2}

, and the counterexample in this paper implies that

t (G) \geq \frac{k + 1}{2}

and

κ (G) \geq k + 2

here cannot be replaced by

t (G) \geq \frac{k + 1}{3}

and

κ (G) \geq k + 1

. However, the bound of toughness is not tight, and it may find a better toughness bound between

\frac{k + 1}{2}

and

\frac{k + 1}{3}

. Zhou 17 determined that a graph

G

is a

(P_{\geq 3}, m)

‐factor deleted graph if

κ (G) \geq 2 m + 1

and

b i n d (G) \geq \frac{6 m + 5}{4 m + 4}

where

m

is a nonnegative number, and the example provided by author showed that the bounds cannot be reduced to

κ (G) \geq 2 m

and

b i n d (G) \geq \frac{6 m + 5}{4 m + 5}

. Although

κ (G)

bound is tight, this example cannot interpret the binding number bound is sharp.…”

Section: Introductionmentioning

confidence: 86%

“…By means of example showed in Zhou, 17 when

b i n d (G)

reach to

\frac{k + 5}{5}

there are graphs are not

(P_{\geq 3}, k)

‐factor critical graphs. Thus

b i n d (G)

condition in Theorem 6 is sharp for

k \geq 3

, and we only need to show

b i n d (G) > \frac{13}{9}

is tight for

k = 2

and

b i n d (G) > \frac{4}{3}

is best for

k = 1

.…”

Section: Sharpnessmentioning

confidence: 99%

“…The third parameter to test the vulnerability of network is binding number which was first introduced by Woodall, 16 while the motivation to introduce this variant is to discuss Anderson number. For a graph

G

, its binding number is formulated by

b i n d (G) = \min (}{, \frac{∣ N_{G} (X) ∣}{∣ X ∣} ∣ \emptyset \neq X \subseteq V (G), N_{G} (X) \neq V (G)) .

It measures the robustness of the network from a neighborhood relation perspective, and is also proved to have strong connection with the existence of factor and fractional factors (see Zhou 17,18 ). Combining with toughness and isolated toughness, these three parameters play an irreplaceable role in the vulnerability of the network and the feasibility of data transmission, and are important factors to be considered in the network design process by the computer scientists.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Tight bounds for the existence of path factors in network vulnerability parameter settings

2020

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The issues of ruggedness and vulnerability are cruxes in network security research, which must be considered during the network designing phase. Parameters such as toughness, isolated toughness, and binding number characterize the vulnerable of the network from the structure of networks. The path factor, a special case of the generalized MJX-tex-caligraphicnormalℋ‐factor, measures the feasibility of data transmission in networks. Recent advances have been obtained to show that there is an inevitable connection between the vulnerability parameters of the network and the existence of path factors, while we found that some existing theoretical results are not tight and there is still a long way for further improvement. In view of graph theory approaches, this paper mainly contributes to determine the sharp bounds of toughness, isolated toughness, and binding number for the existence of path factor in different settings, and therefore solve the open problems left unsolved in previous articles.

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“…A graph

G κ (G) \geq 2 m + 1

and

b i n d (G) > \frac{3 m + 1}{2 m + 1}

is a

(P_{\geq 3}, m)

‐factor deleted graph, where

m \in double-struckN

. The bound of binding number improves the given theorem in Zhou 17 …”

Section: Conclusion and Discussionsupporting

confidence: 51%

“… Zhou et al 22 proved that a graph

G

with

κ (G) \geq k + 2

is a

(P_{\geq 3}, k)

‐factor critical graph if

t (G) \geq \frac{k + 1}{2}

, and the counterexample in this paper implies that

t (G) \geq \frac{k + 1}{2}

and

κ (G) \geq k + 2

here cannot be replaced by

t (G) \geq \frac{k + 1}{3}

and

κ (G) \geq k + 1

. However, the bound of toughness is not tight, and it may find a better toughness bound between

\frac{k + 1}{2}

and

\frac{k + 1}{3}

. Zhou 17 determined that a graph

G

is a

(P_{\geq 3}, m)

‐factor deleted graph if

κ (G) \geq 2 m + 1

and

b i n d (G) \geq \frac{6 m + 5}{4 m + 4}

where

m

is a nonnegative number, and the example provided by author showed that the bounds cannot be reduced to

κ (G) \geq 2 m

and

b i n d (G) \geq \frac{6 m + 5}{4 m + 5}

. Although

κ (G)

bound is tight, this example cannot interpret the binding number bound is sharp.…”

Section: Introductionmentioning

confidence: 86%

“…By means of example showed in Zhou, 17 when

b i n d (G)

reach to

\frac{k + 5}{5}

there are graphs are not

(P_{\geq 3}, k)

‐factor critical graphs. Thus

b i n d (G)

condition in Theorem 6 is sharp for

k \geq 3

, and we only need to show

b i n d (G) > \frac{13}{9}

is tight for

k = 2

and

b i n d (G) > \frac{4}{3}

is best for

k = 1

.…”

Section: Sharpnessmentioning

confidence: 99%

G

, its binding number is formulated by

b i n d (G) = \min (}{, \frac{∣ N_{G} (X) ∣}{∣ X ∣} ∣ \emptyset \neq X \subseteq V (G), N_{G} (X) \neq V (G)) .

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tight bounds for the existence of path factors in network vulnerability parameter settings

2020

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show abstract

“…Zhou, Yang and Xu [22] proved that an n-connected graph G is (P ≥3 , n)-factor critical if its toughness tough(G) ≥ n+1 2 . Some other results on path factors can be found in [3,15,17,18]. Lots of authors derived some toughness conditions for the existence of graph factors [4,5,9,20].…”

Section: Introductionmentioning

confidence: 99%

Some results on path-factor critical avoidable graphs

Zhou¹

2020

Discuss. Math. Graph Theory

Self Cite

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A path factor is a spanning subgraph F of G such that every component of F is a path with at least two vertices. We write P ≥k = {P i : i ≥ k}. Then a P ≥k-factor of G means a path factor in which every component admits at least k vertices, where k ≥ 2 is an integer. A graph G is called a P ≥k-factor avoidable graph if for any e ∈ E(G), G admits a P ≥k-factor excluding e. A graph G is called a (P ≥k , n)-factor critical avoidable graph if for any Q ⊆ V (G) with |Q| = n, G − Q is a P ≥k-factor avoidable graph. Let G be an (n + 2)-connected graph. In this paper, we demonstrate that (i) G is a (P ≥2 , n)-factor critical avoidable graph if tough(G) > n+2 4 ; (ii) G is a (P ≥3 , n)-factor critical avoidable graph if tough(G) > n+1 2 ; (iii) G is a (P ≥2 , n)-factor critical avoidable graph if I(G) > n+2 3 ; (iv) G is a (P ≥3 , n)factor critical avoidable graph if I(G) > n+3 2. Furthermore, we claim that these conditions are sharp.

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Vulnerability Variants and Matching in Networks

Lan

Gao

2020

Machine Learning for Cyber Security

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Remarks on path factors in graphs

Cited by 35 publications

References 24 publications

Tight bounds for the existence of path factors in network vulnerability parameter settings

Tight bounds for the existence of path factors in network vulnerability parameter settings

Some results on path-factor critical avoidable graphs

Vulnerability Variants and Matching in Networks

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