On the Independent Set Sequence of a Tree

Basit, Abdul; Galvin, David

doi:10.37236/9896

Cited by 6 publications

(4 citation statements)

References 17 publications

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“…The proof of Theorem 1.3 gives that one can take c 1 := 0.2403 and c 2 := 2.1243. The existence of the constant c 1 was recently also proved by Basit and Galvin [5]. This "gap" in the values of m for which Tuza's conjecture holds is due to the "wastefulness" of the previous iteration of the online triangle packing process.…”

Section: Introductionmentioning

confidence: 83%

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Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Bennett¹,

Cushman²,

Dudek³

2020

Preprint

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A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number τ (G) is at most twice the triangle packing number ν(G), where the triangle packing number ν(G) is the maximum size of a set of edge-disjoint triangles in G and the triangle covering number τ (G) is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erdős-Rényi random graph G(n, m) for all range of m, closing the "gap" in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and Jinyoung Park.) We employ a random greedy process called the online triangle packing process to produce a triangle packing in G(n, m) and analyze this process by using the differential equations method.

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Section: Introductionmentioning

confidence: 83%

“…Also recall that f 2 ≤ n −1/10 f = O(f ). Finally, observe that by (5), line ( 15) is q b,c . Therefore,…”

Section: 4mentioning

confidence: 93%

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Bennett¹,

Cushman²,

Dudek³

2020

Preprint

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“…where β is the expected size of the largest independent set in G(n, d/n). [BG20] are working on a result similar to Theorem 1.17, and they can also improve on the result of Levit and Mandrescu, Theorem 1.8. 1.4.…”

Section: Theorem 114 ([Cs07]mentioning

confidence: 99%

Independent Sets of Random Trees and of Sparse Random Graphs

Heilman

2020

Preprint

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An independent set of size k in a finite undirected graph G is a set of k vertices of the graph, no two of which are connected by an edge. Let x k (G) be the number of independent sets of size k in the graph G and let α(G) = max{k ≥ 0 : x k (G) = 0}. In 1987, Alavi, Malde, Schwenk and Erdös asked if the independent set sequence x 0 (G), x 1 (G), . . . , x α(G) (G) of a tree is unimodal (the sequence goes up and then down). This problem is still open. In 2006, Levit and Mandrescu showed that the last third of the independent set sequence of a tree is decreasing. We show that the first 46.8% of the independent set sequence of a random tree is increasing with (exponentially) high probability as the number of vertices goes to infinity. So, the question of Alavi, Malde, Schwenk and Erdös is "four-fifths true", with high probability.We also show unimodality of the independent set sequence of Erdös-Renyi random graphs, when the expected degree of a single vertex is large (with (exponentially) high probability as the number of vertices in the graph goes to infinity, except for a small region near the mode). A weaker result is shown for random regular graphs.The structure of independent sets of size k as k varies is of interest in probability, statistical physics, combinatorics, and computer science.

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“…ðÞ , is the independence polynomial of G [1], the independent set polynomial of G [2], or the stable set polynomial of G [3]. Some updated observations concerning the independence polynomial may be found in [4,5].…”

mentioning

confidence: 99%

On Computing of Independence Polynomials of Trees

Ohr¹,

Levit²,

Ron³

et al. 2023

Recent Research in Polynomials [Working Title]

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An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x) = \sum_{k=0}^{\alpha(G)}{s_k}x^{k}={s_0}+{s_1}x+{s_2}x^{2}+...+{s_{\alpha(G)}}x^{\alpha(G)}, \] where $s_k$ denotes the number of independent sets of cardinality $k$ in the graph $G$. A comprehensive survey on the subject is due to Levit and Mandrescu, where some recursive formulas are allowing computation of the independence polynomial. A direct implementation of these recursions does not bring about an efficient algorithm. In 2021, Yosef, Mizrachi, and Kadrawi developed an efficient way for computing the independence polynomials of trees with $n$ vertices, such that a database containing all of the independence polynomials of all the trees with up to $n-1$ vertices is required. This approach is not suitable for big trees, as an extensive database is needed. On the other hand, using dynamic programming, it is possible to develop an efficient algorithm that prevents repeated calculations. In summary, our dynamic programming algorithm runs over a tree in linear time and does not depend on a database.

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On the Independent Set Sequence of a Tree

Cited by 6 publications

References 17 publications

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

Independent Sets of Random Trees and of Sparse Random Graphs

On Computing of Independence Polynomials of Trees

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