Graphs with constant links and induced Turán numbers
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866916401311121408 |
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| author | Caro, Yair Hansberg, Adriana Tuza, Zsolt |
| author_facet | Caro, Yair Hansberg, Adriana Tuza, Zsolt |
| contents | A graph $G$ of constant link $L$ is a graph in which the neighborhood of any vertex induces a graph isomorphic to $L$. Given two different graphs, $H$ and $G$, the induced Turán number ${\rm ex}(n; H, G{\rm -ind})$ is defined as the maximum number of edges in an $n$-vertex graph having no subgraph isomorphic to $H$ and no copy from $G$ as an induced subgraph.
Our main motivation in this paper is to establish a bridge between graphs with constant link and induced Turán numbers via the class of $t$-regular, $k$-uniform (linear) hypergraphs of girth at least $4$, as well as to present several methods of constructing connected graphs with constant link.
We show that, for integers $t \geq 3$ and $k \geq 3$, ${\rm ex}(n; C_k, K_{1,t}{\rm -ind}) \leq (k - 2)(t - 1)n/2$ and that equality holds for infinitely many values of $n$. This result is built upon the existence of graphs with constant link $tL$ with restricted cycle length, which we prove in another theorem. More precisely, we show that, given a graph $F$ with constant link $L$ and circumference $c$, then, for all integers $t \geq 2$ and $g > c$, there exists a graph with constant link $tL$ which is free of cycles of length $l$, for all $c < l < g$. We provide two proofs of this result using distinct approaches. We further present constructions of graphs with constant links $tL$, $t \geq 2$, and restricted cycle length based on Steiner Systems. Finally, starting from a connected graph of constant link $tL$, for $t \geq 2$, having order $n$ and restricted cycle lengths, we provide a method to construct an infinite collection of connected graphs of constant link $tL$ that preserves the cycle length restriction, and whose orders form an arithmetic progression $qn$, $q \geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_12875 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Graphs with constant links and induced Turán numbers Caro, Yair Hansberg, Adriana Tuza, Zsolt Combinatorics 05B05, 05B25, 05C35, 05C65 A graph $G$ of constant link $L$ is a graph in which the neighborhood of any vertex induces a graph isomorphic to $L$. Given two different graphs, $H$ and $G$, the induced Turán number ${\rm ex}(n; H, G{\rm -ind})$ is defined as the maximum number of edges in an $n$-vertex graph having no subgraph isomorphic to $H$ and no copy from $G$ as an induced subgraph. Our main motivation in this paper is to establish a bridge between graphs with constant link and induced Turán numbers via the class of $t$-regular, $k$-uniform (linear) hypergraphs of girth at least $4$, as well as to present several methods of constructing connected graphs with constant link. We show that, for integers $t \geq 3$ and $k \geq 3$, ${\rm ex}(n; C_k, K_{1,t}{\rm -ind}) \leq (k - 2)(t - 1)n/2$ and that equality holds for infinitely many values of $n$. This result is built upon the existence of graphs with constant link $tL$ with restricted cycle length, which we prove in another theorem. More precisely, we show that, given a graph $F$ with constant link $L$ and circumference $c$, then, for all integers $t \geq 2$ and $g > c$, there exists a graph with constant link $tL$ which is free of cycles of length $l$, for all $c < l < g$. We provide two proofs of this result using distinct approaches. We further present constructions of graphs with constant links $tL$, $t \geq 2$, and restricted cycle length based on Steiner Systems. Finally, starting from a connected graph of constant link $tL$, for $t \geq 2$, having order $n$ and restricted cycle lengths, we provide a method to construct an infinite collection of connected graphs of constant link $tL$ that preserves the cycle length restriction, and whose orders form an arithmetic progression $qn$, $q \geq 1$. |
| title | Graphs with constant links and induced Turán numbers |
| topic | Combinatorics 05B05, 05B25, 05C35, 05C65 |
| url | https://arxiv.org/abs/2409.12875 |