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• W3042915511 abstract "This paper considers two important questions in the well-studied theory of graphs that are $F$-saturated. A graph $G$ is called $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$, but the addition of any edge creates a copy of $F$. We first resolve a fundamental question of minimizing the number of cliques of size $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, confirming a conjecture of Kritschgau, Methuku, Tait, and Timmons. We also go further and prove a corresponding stability result. Next we minimize the number of cycles of length $r$ in a $K_s$-saturated graph for all sufficiently large numbers of vertices, and classify the extremal graphs for most values of $r$, answering another question of Kritschgau, Methuku, Tait, and Timmons for most $r$. We then move on to a central and longstanding conjecture in graph saturation made by Tuza, which states that for every graph $F$, the limit $lim_{n rightarrow infty} frac{sat(n, F)}{n}$ exists, where $sat(n, F)$ denotes the minimum number of edges in an $n$-vertex $F$-saturated graph. Pikhurko made progress in the negative direction by considering families of graphs instead of a single graph, and proved that there exists a graph family $mathcal{F}$ of size $4$ for which $lim_{n rightarrow infty} frac{sat(n, mathcal{F})}{n}$ does not exist (for a family of graphs $mathcal{F}$, a graph $G$ is called $mathcal{F}$-saturated if $G$ does not contain a copy of any graph in $mathcal{F}$, but the addition of any edge creates a copy of a graph in $mathcal{F}$, and $sat(n, mathcal{F})$ is defined similarly). We make the first improvement in 15 years by showing that there exist infinitely many graph families of size $3$ where this limit does not exist. Our construction also extends to the generalized saturation problem when we minimize the number of fixed-size cliques." @default.
• W3042915511 created "2020-07-23" @default.
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• W3042915511 date "2020-12-01" @default.
• W3042915511 modified "2023-09-26" @default.
• W3042915511 title "Minimizing the numbers of cliques and cycles of fixed size in an <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML display=inline id=d1e22 altimg=si31.svg><mml:mi>F</mml:mi></mml:math>-saturated graph" @default.
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• W3042915511 doi "https://doi.org/10.1016/j.ejc.2020.103185" @default.
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