Spectral extremal problem on $t$ copies of $\ell$-cycle

Longfei, Fang; Mingqing, Zhai; Lin, Huiqiu

doi:10.48550/arxiv.2302.03229

Cited by 3 publications

(5 citation statements)

References 30 publications

(54 reference statements)

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“…The behavior for degenerate hypergraphs seems very different from nondegenerate hypergraphs, and we refer the reader to e.g. [18,Theorem 1.3] for related results on even cycles.…”

Section: Discussionmentioning

confidence: 99%

A step towards a general density Corrádi--Hajnal Theorem

Hou¹,

Li²,

Liu³

et al. 2023

Preprint

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For a nondegenerate r-graph F , large n, and t in the regime [0, c F n], where c F > 0 is a constant depending only on F , we present a general approach for determining the maximum number of edges in an n-vertex r-graph that does not contain t + 1 vertex-disjoint copies of F . In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions.Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Simonovits [65] and Moon [53] on complete graphs and can be viewed as a step towards a general density version of the classical Corrádi-Hajnal Theorem [10].

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“…The behavior for degenerate hypergraphs seems very different from nondegenerate hypergraphs, and we refer the reader to e.g. [18,Theorem 1.3] for related results on even cycles.…”

Section: Discussionmentioning

confidence: 99%

A step towards a general density Corrádi--Hajnal Theorem

Hou¹,

Li²,

Liu³

et al. 2023

Preprint

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“…Let Ex sp (n, F ) denote the set of all extremal graphs with respect to ex sp (n, F ). This problem was also studied extensively: see complete graph [3,34], odd cycle [24], 4-cycle [15,23,36], book graph [35], odd wheel graph [10], theta graph [35], complete bipartite graph [1,27], path [28], friendship graph [11] and some nice surveys [8,17,19,20,21,29]. Clearly, book graph, theta graph, complete graph and odd cycle are all color-critical graphs.…”

Section: Theorem 1 ([32]mentioning

confidence: 99%

“…, F t . This topic was studied for some special cases: matching [14], complete graph [30], star [7], path [6], odd cycle [15], even cycle [15]. Let K + r (a 1 , a 2 , .…”

Section: Theorem 2 ([25]mentioning

confidence: 99%

“…The well-known Turán Theorem shows that T n,r is the extremal graph corresponding to ex(n, K r+1 ). Later, Moon [22] and Simonovits [31] showed that K t−1 ∨ T n−t+1,r is the unique extremal graph corresponding to ex(n, tK r+1 ) for sufficiently large n. Recently, Fang et al [15] determined the unique extremal graph with respect to ex(n, tC 2r+1 ). For more advances on this topic, we refer the readers to [4,13,16,18,33].…”

Section: Introductionmentioning

confidence: 99%

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Spectral Extremal Problem on Disjoint Color-Critical Graphs

Lei,

2024

Electron. J. Combin.

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For a given graph $F$, we say that a graph $G$ is $F$-free if it does not contain $F$ as a subgraph. A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. Let $K_r^{+}(a_1, a_2, \ldots, a_r)$ be the graph obtained from complete $r$-partite graph with parts of sizes $a_1\geqslant 2, a_2, \ldots, a_r$, by adding an edge to the first part. In this paper, we focus on the spectral extrema of disjoint color-critical graphs. For fixed $t, a_1,\ldots, a_r\ (r\geqslant 2)$ and large enough $n$, we characterize the unique $n$-vertex $tK_r^+(a_1,\ldots,a_r)$-free graph having the largest spectral radius. Moreover, let $F_1,\ldots,F_t$ be $t$ disjoint color-critical graphs with the same chromatic number. We identify the unique $n$-vertex $\bigcup_{i=1}^tF_i$-free graph having the largest spectral radius for sufficiently large $n$. Consequently, we generalize the main results obtained by Ni, Wang and Kang [Electron. J. Combin. 30 (1) (2023), #P1.20] and by Fang, Zhai and Lin [arXiv:2302.03229v2].

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“…In the past decades, the Brualdi-Soheid-Turán type problem has been studied by many researchers for many specific graphs, such as complete graphs [18,24], odd cycles [19], even cycles [2,18,26,27], paths [20] and wheels [3,28]. For more information, we refer the reader to [4,9,12,13,14,15,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Spectral Extremal Results on Trees

Fang,

Lin,

Shu

et al. 2024

Electron. J. Combin.

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Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where '$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

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Spectral extremal problem on $t$ copies of $\ell$-cycle

Cited by 3 publications

References 30 publications

A step towards a general density Corrádi--Hajnal Theorem

A step towards a general density Corrádi--Hajnal Theorem

Spectral Extremal Problem on Disjoint Color-Critical Graphs

Spectral Extremal Results on Trees

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