All minimum C5-saturated graphs

Chen, Ya‐Chen

doi:10.1002/jgt.20508

Cited by 22 publications

(26 citation statements)

References 18 publications

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“…The case of

sat (n, C_{3}) = n - 1

is trivial; the cases

k = 4

and

k = 5

were established by Ollmann in 1972 and by Ya‐Chen in 2009, respectively.

\begin{matrix} true \\ right sat (n, C_{4}) & = & left (⌊⌋, \frac{3 n - 5}{2}) for n \geq 5 . \\ right sat (n, C_{5}) & = & left (⌈⌉, \frac{10 (n - 1)}{7}) for n \geq 21 . \end{matrix}

Actually, was conjectured by Fisher, Fraughnaugh, Langley .…”

Section: Cycle‐saturated Graphsmentioning

confidence: 95%

“…Actually, (2) was conjectured by Fisher, Fraughnaugh, Langley [13]. Later Ya-Chen [8] determined sat(n, C 5 ) for all n, as well as all extremal graphs. The best previously known general lower bound came from Barefoot, Clark, Entringer, Porter, Székely, and Tuza [3], and the best general upper bound (a clever, complicated construction resembling a bicycle wheel) came from Gould, Łuczak, and Schmitt [15] 1…”

Section: Theorem 21mentioning

confidence: 97%

“…

\begin{matrix} true \\ right sat (n, C_{4}) & = & left (⌊⌋, \frac{3 n - 5}{2}) for n \geq 5 . \\ right sat (n, C_{5}) & = & left (⌈⌉, \frac{10 (n - 1)}{7}) for n \geq 21 . \end{matrix}

Actually, was conjectured by Fisher, Fraughnaugh, Langley . Later Ya‐Chen determined

sat (n, C_{5})

for all n , as well as all extremal graphs.…”

Section: Cycle‐saturated Graphsmentioning

confidence: 99%

See 2 more Smart Citations

Cycle-Saturated Graphs with Minimum Number of Edges

Füredi

Kim

2012

J. Graph Theory

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A graph G is called H‐saturated if it does not contain any copy of H, but for any edge e in the complement of G, the graph G+e contains some H. The minimum size of an n‐vertex H‐saturated graph is denoted by sat(n,H). We prove sat(n,Ck)=n+n/k+O((n/k2)+k2)holds for all n≥k≥3, where Ck is a cycle with length k. A graph G is H‐semisaturated if G+e contains more copies of H than G does for ∀e∈E(G¯). Let ssat (n,H) be the minimum size of an n‐vertex H‐semisaturated graph. We have ssat(n,Ck)=n+n/(2k)+O((n/k2)+k).We conjecture that our constructions are optimal for n>n0(k). © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 203–215, 2013

show abstract

“…The case of

sat (n, C_{3}) = n - 1

is trivial; the cases

k = 4

and

k = 5

were established by Ollmann in 1972 and by Ya‐Chen in 2009, respectively.

\begin{matrix} true \\ right sat (n, C_{4}) & = & left (⌊⌋, \frac{3 n - 5}{2}) for n \geq 5 . \\ right sat (n, C_{5}) & = & left (⌈⌉, \frac{10 (n - 1)}{7}) for n \geq 21 . \end{matrix}

Actually, was conjectured by Fisher, Fraughnaugh, Langley .…”

Section: Cycle‐saturated Graphsmentioning

confidence: 95%

Section: Theorem 21mentioning

confidence: 97%

See 1 more Smart Citation

Cycle-Saturated Graphs with Minimum Number of Edges

Füredi

Kim

2012

J. Graph Theory

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show abstract

“…There are very few graphs for which sat(H, n) is known exactly. In addition to cliques, sat(H, n) is known for stars, paths and matchings [12], tK p , K p ∪ K q and generalized friendship graphs [8], books [3,16], C 4 [15], C 5 [4], and K 2,3 [17]. Some progress has been made for arbitrary cycles, and the current best known upper bound on sat(C t , n) can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

The game of F-saturator

Ferrara

Jacobson

Harris

2010

Discrete Applied Mathematics

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“…(A shorter proof was found by Tuza [29].) Very recently, Chen [9,8] completely solved the case of C 5 . The value of sat(n, C m ) is unknown (even asymptotically) for any other fixed m; various bounds are proved in [1,17].…”

mentioning

confidence: 98%