Induced saturation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si18.svg"><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:math>

Raty, Eero

doi:10.1016/j.disc.2019.111641

Cited by 3 publications

(1 citation statement)

References 4 publications

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“…This leads to a question, asked by Axenovich and Csikós [1], for what integers n 5 do there exist P n -IS graphs. Räty [7] was the first to make a progress on this question, showing by an algebraic construction that there exists a P 6 -IS graph. Cho, Choi and Park [4] later showed that in fact for any k 2, there exists a P 3k -IS graph.…”

Section: Introductionmentioning

confidence: 99%

$P_{n}$-Induced-Saturated Graphs Exist for all $n \geqslant 6$

Dvořák

2020

Electron. J. Combin.

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Let $P_{n}$ be a path graph on $n$ vertices. We say that a graph $G$ is $P_{n}$-induced-saturated if $G$ contains no induced copy of $P_{n}$, but deleting any edge of $G$ as well as adding to $G$ any edge of $G^{c}$ creates such a copy. Martin and Smith (2012) showed that there is no $P_{4}$-induced-saturated graph. On the other hand, there trivially exist $P_{n}$-induced-saturated graphs for $n=2,3$. Axenovich and Csikós (2019) ask for which integers $n \geqslant 5$ do there exist $P_{n}$-induced-saturated graphs. Räty (2019) constructed such a graph for $n=6$, and Cho, Choi and Park (2019) later constructed such graphs for all $n=3k$ for $k \geqslant 2$. We show by a different construction that $P_{n}$-induced-saturated graphs exist for all $n \geqslant 6$, leaving only the case $n=5$ open.

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Section: Introductionmentioning

confidence: 99%

$P_{n}$-Induced-Saturated Graphs Exist for all $n \geqslant 6$

Dvořák

2020

Electron. J. Combin.

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On induced saturation for paths

Cho

Choi

Park

2021

European Journal of Combinatorics

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On induced saturation for paths

Cho

Choi

Park

2019

Preprint

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For a graph H, a graph G is H-induced-saturated if G does not contain an induced copy of H, but either removing an edge from G or adding a non-edge to G creates an induced copy of H. Depending on the graph H, an H-induced-saturated graph does not necessarily exist. In fact, Martin and Smith [11] showed that P 4 -induced-saturated graphs do not exist, where P k denotes a path on k vertices. Axenovich and Csikós [1] asked the existence of P k -induced-saturated graphs for k ≥ 5; it is easy to construct such graphs when k ∈ {2, 3}. Recently, Räty [13] constructed a graph that is P 6 -induced-saturated.In this paper, we show that there exists a P k -induced-saturated graph for infinitely many values of k. To be precise, we find a P 3n -induced-saturated graph for every positive integer n. As a consequence, for each positive integer n, we construct infinitely many P 3n -induced-saturated graphs. We also show that the Kneser graph K(n, 2) is P 6 -induced-saturated for every n ≥ 5.

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Induced saturation of P6

Cited by 3 publications

References 4 publications

$P_{n}$-Induced-Saturated Graphs Exist for all $n \geqslant 6$

$P_{n}$-Induced-Saturated Graphs Exist for all $n \geqslant 6$

On induced saturation for paths

On induced saturation for paths

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