Cycle-saturated graphs of minimum size

Barefoot, C. A.; Clark, Lane H.; Entringer, R. C.; Porter, Thomas Dale; Székely, László Á.; Tuza, Zs.

doi:10.1016/0012-365x(95)00173-t

Cited by 22 publications

(48 citation statements)

References 8 publications

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“…where ε(k) = 2 for k even ≥ 10, ε(k) = 3 for k odd ≥ 17. Although there is still a gap, Theorem 2.1 supersedes all earlier results for k ≥ 6 except the construction giving sat(n, C 6 ) ≤ 3 2 n for n ≥ 11 from [3]. Our new construction for a k-cycle-saturated graph for n = (k − 1) + t(k − 4), where k ≥ 7, t ≥ 1, can be read from the picture below.…”

Section: Theorem 21mentioning

confidence: 52%

“…Thus, divide

V (G)

into five parts

{X, Y_{3}, Y_{4 +}, Z_{2}, Z_{3 +}}

\begin{matrix} true \\ right Y_{3} : = {v \in Y : {prefixdeg}_{G} (v) = 3} and Y_{4 +} : = {v \in Y : {prefixdeg}_{G} (v) \geq 4}, \\ right Z_{2} : = {v \in Z : {prefixdeg}_{G} (v) = 2} and Z_{3 +} : = {v \in Z : {prefixdeg}_{G} (v) \geq 3} . \end{matrix}

Lemma ( The structure of

C_{k}

‐ saturated graphs . See ). Suppose that G is a

C_{k}

‐ saturated graph ( and

k \geq 5

).…”

Section: Degree One Vertices In (Semi)saturated Graphsmentioning

confidence: 99%

“…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: 99%

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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

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Section: Theorem 21mentioning

confidence: 52%

“…Thus, divide

V (G)

into five parts

{X, Y_{3}, Y_{4 +}, Z_{2}, Z_{3 +}}

\begin{matrix} true \\ right Y_{3} : = {v \in Y : {prefixdeg}_{G} (v) = 3} and Y_{4 +} : = {v \in Y : {prefixdeg}_{G} (v) \geq 4}, \\ right Z_{2} : = {v \in Z : {prefixdeg}_{G} (v) = 2} and Z_{3 +} : = {v \in Z : {prefixdeg}_{G} (v) \geq 3} . \end{matrix}

Lemma ( The structure of

C_{k}

‐ saturated graphs . See ). Suppose that G is a

C_{k}

‐ saturated graph ( and

k \geq 5

).…”

Section: Degree One Vertices In (Semi)saturated Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Cycle-Saturated Graphs with Minimum Number of Edges

Füredi

Kim

2012

J. Graph Theory

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“…Barefoot et al [1] gave an asymptotic value of mx(n, C k ). They proved that for any natural number k and sufficiently large n, n + an k mx(n, C k ) n + bn k for some positive constants a and b.…”

Section: Introductionmentioning

confidence: 98%

C3 saturated graphs

Ashkenazi¹

2005

Discrete Mathematics

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The first part of this paper deals with the properties of C 3 -saturated graphs. It will be shown that for any C 3 -saturated graph, G,with which two vertices are adjacent iff the distance between them, in G, is k. In addition to this, a full description of the set of planar C 3 -saturated graphs, PSAT(n, C 3 ), will be given. It will be shown that there are only three kinds of such graphs. In the second part of the paper a useful characterization of graphs which are C 3 -saturated and C 4 -free will be given in terms of the adjacency and incidence matrices A and B.

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“…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]. Also, the aggregate outcome of papers [6,19,11,10,12,32], with the final gaps filled by computer search, determines sat(n, C n ), the Hamilton cycle case, for all n ≥ 3.…”

mentioning

confidence: 99%