Saturation numbers for families of Ramsey-minimal graphs

Chen, Guantao; Ferrara, Michael; Gould, Ronald J.; Magnant, Colton; Schmitt, John R.

doi:10.4310/joc.2011.v2.n3.a5

Cited by 10 publications

(14 citation statements)

References 7 publications

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“…In [2], Chen et al verified the Hanson-Toft conjecture for sat(n, R min (K 3 , K 3 )), the first nontrivial case. They also proved an upper bound on sat(n, R min (K t , T m )) where T m is a tree of order m and determined sat(n, R min (K 3 , P 3 )).…”

Section: Conjecturementioning

confidence: 97%

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Colored Saturation Parameters for Rainbow Subgraphs

Barrus

Ferrara

Vandenbussche

et al. 2017

Journal of Graph Theory

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Inspired by a 1987 result of Hanson and Toft [Edge‐colored saturated graphs, J Graph Theory 11 (1987), 191–196] and several recent results, we consider the following saturation problem for edge‐colored graphs. An edge‐coloring of a graph F is rainbow if every edge of F receives a different color. Let R(F) denote the set of rainbow‐colored copies of F. A t‐edge‐colored graph G is (R(F),t)‐saturated if G does not contain a rainbow copy of F but for any edge e∈E(G¯) and any color i∈[t], the addition of e to G in color i creates a rainbow copy of F. Let sat tfalse(n,frakturR(F)false) denote the minimum number of edges in an (R(F),t)‐saturated graph of order n. We call this the rainbow saturation number of F. In this article, we prove several results about rainbow saturation numbers of graphs. In stark contrast with the related problem for monochromatic subgraphs, wherein the saturation is always linear in n, we prove that rainbow saturation numbers have a variety of different orders of growth. For instance, the rainbow saturation number of the complete graph Kn lies between nlogn/loglogn and nlogn, the rainbow saturation number of an n‐vertex star is quadratic in n, and the rainbow saturation number of any tree that is not a star is at most linear.

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Section: Conjecturementioning

confidence: 97%

“…In , Chen et al. verified the Hanson–Toft conjecture for

sat (n, {scriptR}_{\min} false(K_{3}, K_{3} false))

, the first nontrivial case.…”

Section: Introductionmentioning

confidence: 98%

Colored Saturation Parameters for Rainbow Subgraphs

Barrus

Ferrara

Vandenbussche

et al. 2017

Journal of Graph Theory

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“…We may assume that d b (y) = 1. By (5), yz / ∈ E(G). Let y 1 ∈ C be the unique neighbor of y in G b , and let z 1 ∈ B be a neighbor of z in G b .…”

Section: K 3 -Saturated Graphsmentioning

confidence: 99%

“…. , K kt )) = n 2 n < r (r − 2)(n − r + 2) + r−2 2 n ≥ r Chen, Ferrara, Gould, Magnant, and Schmitt [5] proved that sat(n, R min (K 3 , K 3 )) = 4n − 10 for n ≥ 56. This settles the first non-trivial case of Conjecture 1.1 for sufficiently large n, and is so far the only settled case.…”

Section: Introductionmentioning

confidence: 99%

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Saturation numbers for Ramsey-minimal graphs

Rolek

Song

2018

Discrete Mathematics

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Given graphs H 1 , . . . , H t , a graph G is (H 1 , . . . , H t )-Ramsey-minimal if every tcoloring of the edges of G contains a monochromatic H i in color i for some i ∈ {1, . . . , t}, but any proper subgraph of G does not possess this property. We define R min (H 1 , . . . , H t ) to be the family of (H 1 , . . . ,We define sat(n, R min (H 1 , . . . , H t )) to be the minimum number of edges over all R min (H 1 , . . . , H t )-saturated graphs on n vertices. In 1987, Hanson and Toft conjectured that sat(n, R min (K k 1 , . . . , K kt )) = (r − 2)(n − r + 2) + r−2 2 for n ≥ r, where r = r(K k 1 , . . . , K kt ) is the classical Ramsey number for complete graphs. The first non-trivial case of Hanson and Toft's conjecture for sufficiently large n was setteled in 2011, and is so far the only settled case. Motivated by Hanson and Toft's conjecture, we study the minimum number of edges over all R min (K 3 , T k )-saturated graphs on n vertices, where T k is the family of all trees on k vertices. We show that for n ≥ 18, sat(n, R min (K 3 , T 4 )) = ⌊5n/2⌋. For k ≥ 5 and n ≥ 2k + (⌈k/2⌉ + 1)⌈k/2⌉ − 2, we obtain an asymptotic bound for sat(n, R min (K 3 , T k )) by showing that 3 2 + 1

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