Cores of cubelike graphs

Mančinska, Laura; Pivotto, Irene; Roberson, David E.; Royle, Gordon F.

doi:10.1016/j.ejc.2020.103092

Cited by 3 publications

(2 citation statements)

References 8 publications

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“…In fact, for certain classes of graphs this question translates to some of longstanding open problems in finite geometry [12,97]. For some other results related to cores see [40,41,75,99,105].…”

Section: Introductionmentioning

confidence: 99%

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

Orel¹

2021

Preprint

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Given a finite simple graph Γ on n vertices its complementary prism is the graph Γ Γ that is obtained from Γ and its complement Γ by adding a perfect matching, where each its edge connects two copies of the same vertex in Γ and Γ. It generalizes the Petersen graph, which is obtained if Γ is the pentagon. The automorphism group of Γ Γ is described for arbitrary graph Γ. In particular, it is shown that the ratio between the cardinalities of the automorphism groups of Γ Γ and Γ can attain only values 1, 2, 4, and 12. It is shown that the Cheeger number of Γ Γ equals either 1 or 1 − 1 n , and the two corresponding classes of graphs are fully determined. It is proved that Γ Γ is vertex-transitive if and only if Γ is vertex-transitive and self-complementary. In this case the complementary prism is Hamiltonian-connected whenever n > 5, and is not a Cayley graph whenever n > 1. The main results involve endomorphisms of graph Γ Γ. It is shown that Γ Γ is a core, i.e. all its endomorphisms are automorphisms, whenever Γ is strongly regular and self-complementary. The same conclusion is obtained for many vertex-transitive self-complementary graphs. In particular, it is shown that if there exists a vertex-transitive self-complementary graph Γ such that Γ Γ is not a core, then Γ is neither a core nor its core is a complete graph.

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Section: Introductionmentioning

confidence: 99%

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

Orel¹

2021

Preprint

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“…Our first result extends the lower bound to any Cayley graph of Z n 2 . These graphs are sometimes called cubelike graphs, see, for example, [13]. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Unitary signings and induced subgraphs of Cayley graphs of $\mathbb{Z}_2^{n}$

Alon

Zheng

2020

AiC

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Boolean functions play an important role in many different areas of computer science. The _local sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of coordinates whose flip changes the value of $f(x)$, i.e., the number of i's such that $f(x)\not=f(x+e_i)$, where $e_i$ is the $i$-th unit vector. The _sensitivity_ of a Boolean function is its maximum local sensitivity. In other words, the sensitivity measures the robustness of a Boolean function with respect to a perturbation of its input. Another notion that measures the robustness is block sensitivity. The _local block sensitivity_ of a Boolean function $f:\{0,1\}^n\to \{0,1\}$ on an input $x\in\{0,1\}^n$ is the number of disjoint subsets $I$ of $\{1,..,n\}$ such that flipping the coordinates indexed by $I$ changes the value of $f(x)$, and the _block sensitivity_ of $f$ is its maximum local block sensitivity. Since the local block sensitivity is at least the local sensitivity for any input $x$, the block sensitivity of $f$ is at least the sensitivity of $f$. The next example demonstrates that the block sensitivity of a Boolean function is not linearly bounded by its sensitivity. Fix an integer $k\ge 2$ and define a Boolean function $f:\{0,1\}^{2k^2}\to\{0,1\}$ as follows: the coordinates of $x\in\{0,1\}^{2k^2}$ are split into $k$ blocks of size $2k$ each and $f(x)=1$ if and only if at least one of the blocks contains exactly two entries equal to one and these entries are consecutive. While the sensitivity of the function $f$ is $2k$, its block sensitivity is $k^2$. The Sensitivity Conjecture, made by Nisan and Szegedy in 1992, asserts that the block sensitivity of a Boolean function is polynomially bounded by its sensivity. The example above shows that the degree of such a polynomial must be at least two. The Sensitivity Conjecture has been recently proven by Huang in [Annals of Mathematics 190 (2019), 949-955](https://doi.org/10.4007/annals.2019.190.3.6). He proved the following combinatorial statement that implies the conjecture (with the degree of the polynomial equal to four): any subset of more than half of the vertices of the $n$-dimensional cube $\{0,1\}^n$ induces a subgraph that contains a vertex with degree at least $\sqrt{n}$. The present article extends this result as follows: every Cayley graph with the vertex set $\{0,1\}^n$ and any generating set of size $d$ (the vertex set is viewed as a vector space over the binary field) satisfies that any subset of more than half of its vertices induces a subgraph that contains a vertex of degree at least $\sqrt{d}$. In particular, when the generating set consists of the $n$ unit vectors, the Cayley graph is the $n$-dimensional hypercube.

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The core of a complementary prism

Orel

2023

J Algebr Comb

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The complementary prism $$\Gamma {\bar{\Gamma }}$$ Γ Γ ¯ is constructed from the disjoint union of a graph $$\Gamma $$ Γ and its complement $${\bar{\Gamma }}$$ Γ ¯ if an edge is added between each pair of identical vertices in $$\Gamma $$ Γ and $${\bar{\Gamma }}$$ Γ ¯ . It generalizes the Petersen graph, which is obtained if $$\Gamma $$ Γ is the pentagon. The core of the complementary prism is investigated for arbitrary simple graph $$\Gamma $$ Γ . In particular, it is shown that if $$\Gamma $$ Γ is strongly regular and self-complementary, then $$\Gamma {\bar{\Gamma }}$$ Γ Γ ¯ is a core, i.e. all its endomorphisms are automorphisms.

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Cores of cubelike graphs

Cited by 3 publications

References 8 publications

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

Unitary signings and induced subgraphs of Cayley graphs of $\mathbb{Z}_2^{n}$

The core of a complementary prism

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