Saturated Graphs of Prescribed Minimum Degree

Day, A. Nicholas

doi:10.1017/s0963548316000377

Cited by 15 publications

(15 citation statements)

References 12 publications

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“…For larger t and s, the bounds are not exact. In 2014, Day [6], resolving a conjecture of Bollobás [14] from 1996, showed that sat t (n, K s ) ≥ tn − c t (1) for any s ≥ 3, t ≥ s − 2. Here c t is a constant depending only on t. Further discussion on saturated graphs with degree constraints can be found in [10].…”

Section: History and Previous Resultsmentioning

confidence: 99%

Triangles in Ks-saturated graphs with minimum degree t

Timmons¹,

Cole²,

Curry³

et al. 2020

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For n ≥ 15, we prove that the minimum number of triangles in an n-vertex K 4saturated graph with minimum degree 4 is exactly 2n − 4, and that there is a unique extremal graph. This is a triangle version of a result of Alon, Erdős, Holzman, and Krivelevich from 1996. Additionally, we show that for any s > r ≥ 3 and t ≥ 2(s−2)+1, there is a K s-saturated n-vertex graph with minimum degree t that has s−2 r−1 2 r−1 n + c s,r,t copies of K r. This shows that unlike the number of edges, the number of K r 's (r > 2) in a K s-saturated graph is not forced to grow with the minimum degree, except for possibly in lower order terms.

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Section: History and Previous Resultsmentioning

confidence: 99%

Triangles in Ks-saturated graphs with minimum degree t

Timmons¹,

Cole²,

Curry³

et al. 2020

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“…Erdős, Hajnal and Moon [7] in 1964 initiated the study of the minimum number of edges over all K t -saturated graphs on n vertices (see the dynamic survey [8] on the extensive studies on K t -saturated graphs). Theorem 5 below is a result of Day [6] on K t -saturated graphs with prescribed minimum degree. It confirms a conjecture of Bollobás [1] when t = 3.…”

Section: Conjecture 3 (Hanson and Toftmentioning

confidence: 99%

On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

Song

Zhang²

2021

Electron. J. Combin.

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Given an integer $r\geqslant 1$ and graphs $G, H_1, \ldots, H_r$, we write $G \rightarrow ({H}_1, \ldots, {H}_r)$ if every $r$-coloring of the edges of $G$ contains a monochromatic copy of $H_i$ in color $i$ for some $i\in\{1, \ldots, r\}$. A non-complete graph $G$ is $(H_1, \ldots, H_r)$-co-critical if $G \nrightarrow ({H}_1, \ldots, {H}_r)$, but $G+e\rightarrow ({H}_1, \ldots, {H}_r)$ for every edge $e$ in $\overline{G}$. In this paper, motivated by Hanson and Toft's conjecture [Edge-colored saturated graphs, J Graph Theory 11(1987), 191–196], we study the minimum number of edges over all $(K_t, \mathcal{T}_k)$-co-critical graphs on $n$ vertices, where $\mathcal{T}_k$ denotes the family of all trees on $k$ vertices. Following Day [Saturated graphs of prescribed minimum degree, Combin. Probab. Comput. 26 (2017), 201–207], we apply graph bootstrap percolation on a not necessarily $K_t$-saturated graph to prove that for all $t\geqslant4 $ and $k\geqslant\max\{6, t\}$, there exists a constant $c(t, k)$ such that, for all $n \ge (t-1)(k-1)+1$, if $G$ is a $(K_t, \mathcal{T}_k)$-co-critical graph on $n$ vertices, then $$ e(G)\geqslant \left(\frac{4t-9}{2}+\frac{1}{2}\left\lceil \frac{k}{2} \right\rceil\right)n-c(t, k).$$ Furthermore, this linear bound is asymptotically best possible when $t\in\{4,5\}$ and $k\geqslant6$. The method we develop in this paper may shed some light on attacking Hanson and Toft's conjecture.

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“…We obtain a bad 2-coloring of G + y 2 v from c by coloring the edge y 2 v red, and then when t = 2, recoloring the edges yz 1 , z 1 y 1 , z 2 y 2 , y 2 z blue, the edges z 1 z, z 1 z 2 , and all the edges incident with y 2 in G b red; when t = 3, recoloring the edges y 1 z 1 , y 1 z 2 , zy 2 , zy 3 blue, the edges yy 1 , zz 1 , zz 2 , and all the edges between A and {y 2 , y 3 } in G b red. By (7), d b (y) = d b (z) = 2. By (5), yz / ∈ E(G).…”

Section: K 3 -Saturated Graphsmentioning

confidence: 99%

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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Saturated Graphs of Prescribed Minimum Degree

Cited by 15 publications

References 12 publications

Triangles in Ks-saturated graphs with minimum degree t

Triangles in Ks-saturated graphs with minimum degree t

On the Size of $(K_t,\mathcal{T}_k)$-Co-Critical Graphs

Saturation numbers for Ramsey-minimal graphs

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