An Erdős-Gallai type theorem for vertex colored graphs

Salia, Nika; Tompkins, Casey; Zamora, Oscar

doi:10.1007/s00373-019-02026-1

Cited by 3 publications

(1 citation statement)

References 4 publications

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“…The reader interested in the problems and extensions related to Erdős-Sós conjecture we refer to the following papers [20,19,14,23,18], extensions for Berge hypergraphs see [13,15], extensions for colored graphs see [21].…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

How connectivity affects the extremal number of trees

Jiang¹,

Liu²,

Salia³

2023

Preprint

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How connectivity affects the extremal number of trees

Jiang,

Liu,

Salia

2024

Journal of Combinatorial Theory, Series B

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How connectivity affects the extremal number of trees

Jiang

Liu

Salia

2023

Proceedings of the 12th European Conference on Combinatorics, Graph Theory and Applications

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The Erd\H{o}s-Sós conjecture states that the maximum number of edges in an $n$-vertex graph without a given $k$-vertex tree is at most $\frac {n(k-2)}{2}$. 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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An Erdős-Gallai type theorem for vertex colored graphs

Cited by 3 publications

References 4 publications

How connectivity affects the extremal number of trees

How connectivity affects the extremal number of trees

How connectivity affects the extremal number of trees

How connectivity affects the extremal number of trees

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