Extremal graphs for edge blow-up of graphs

Yuan, Long-Tu

doi:10.1016/j.jctb.2021.10.006

Cited by 18 publications

(13 citation statements)

References 19 publications

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“…Some of these graphs (so-called edge blow-up of stars) have been studied before, see e.g. [3,6,24]. As the path is also a broom, the result below can be viewed as an extension of Kopylov's result (1).…”

mentioning

confidence: 88%

How connectivity affects the extremal number of trees

Jiang¹,

Liu²,

Salia³

2023

Preprint

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The Erdős-Sós conjecture states that the maximum number of edges in an n-vertex graph without a given k-vertex tree is at most n(k−2) 2. Despite significant interest, the conjecture remains unsolved. Recently, Caro, Patkós, and Tuza considered this problem for host graphs that are connected. Settling a problem posed by them, for a k-vertex tree T , we construct n-vertex connected graphs that are T -free with at least (1/4 − o k (1))nk edges, showing that the additional connectivity condition can reduce the maximum size by at most a factor of 2. Furthermore, we show that this is optimal: there is a family of k-vertex brooms T such that the maximum size of an n-vertex connected T -free graph is at most (1/4 + o k (1))nk.

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confidence: 88%

How connectivity affects the extremal number of trees

Jiang¹,

Liu²,

Salia³

2023

Preprint

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“…Later, Ma [16] and Liu et al [14] confirmed Conjecture 1 for a large family of bipartite graphs, including cycles and some complete bipartite graphs. Recently, Yuan [20] found some counterexamples to Conjecture 1 with large chromatic number. However, it remains unknown if Conjecture 1 holds for all bipartite graphs or other graphs with small chromatic number.…”

Section: Remark On Conjecturementioning

confidence: 99%

“…Liu [13] also determined the Turán number for the edge blow-up of some trees and all cycles. Recently, Wang et al [18] determined the extremal graphs for the edge blow-up of a large family of trees and Yuan [20] got a tight bound for any G p+1 when p ≥ χ(G).…”

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confidence: 99%

Turán number for odd-ballooning of trees

Xiutao¹,

Chen²

2022

Preprint

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The Turán number ex(n, H) is the maximum number of edges in an H-free graph on n vertices. Let T be any tree. The odd-ballooning of T , denoted by T o , is a graph obtained by replacing each edge of T with an odd cycle containing the edge, and all new vertices of the odd cycles are distinct. In this paper, we determine the exact value of ex(n, T o ) for sufficiently large n and T o being good, which generalizes all the known results on ex(n, T o ) for T being a star, due to Erdős et al. (1995), Hou et al. (2018 and Yuan ( 2018), and provides some counterexamples with chromatic number 3 to a conjecture of Keevash and Sudakov (2004), on the maximum number of edges not in any monochromatic copy of H in a 2-edge-coloring of a complete graph of order n.

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“…The range of Turán numbers for edge blow-up of all bipartite graphs when p ≥ 3 and the exact Turán numbers for edge blow-up of all non-bipartite graphs when p ≥ χ(H) + 1 has been determined by Yuan in [15]. The following Theorem 1.1 (i) and (ii) are implied by Theorem 2.3 in [15] by taking B = {K t } when ℓ is odd, and B = {K t+1 } when ℓ is even; and Theorem 1.1 (iii) and (iv) are implied by Theorem 2.4 in [15] by taking B = {K t } when ℓ is odd, and B = {K t+1 } when ℓ is even.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently some substantial progress of the extremal graphs for H p+1 of larger p has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when p ≥ 3 and the exact Turán numbers for edge blow-up of all non-bipartite graphs when p ≥ χ(H) + 1 has been determined by Yuan (2022), where χ(H) is the chromatic number of H. A lollipop C k, ℓ is the graph obtained from a cycle C k by appending a path P ℓ+1 to one of its vertices. In this paper, we consider the extremal graphs for C p+1 k, ℓ of the rest cases p = 2 and p = 3.…”

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confidence: 99%

Extremal graphs for edge blow-up of lollipops

Zhai¹,

Yuan²,

Ni³

2022

Preprint

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Given a graph H and an integer p (p ≥ 2), the edge blow-up H p+1 of H is the graph obtained from replacing each edge in H by a clique of order (p + 1), where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. Very recently some substantial progress of the extremal graphs for H p+1 of larger p has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when p ≥ 3 and the exact Turán numbers for edge blow-up of all non-bipartite graphs when p ≥ χ(H) + 1 has been determined by Yuan (2022), where χ(H) is the chromatic number of H. A lollipop C k, ℓ is the graph obtained from a cycle C k by appending a path P ℓ+1 to one of its vertices. In this paper, we consider the extremal graphs for C p+1 k, ℓ of the rest cases p = 2 and p = 3.

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Extremal graphs for edge blow-up of graphs

Cited by 18 publications

References 19 publications

How connectivity affects the extremal number of trees

How connectivity affects the extremal number of trees

Turán number for odd-ballooning of trees

Extremal graphs for edge blow-up of lollipops

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