Discrete Mathematics最新文献

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Hamiltonicity of certain vertex-transitive graphs revisited

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-10 DOI: 10.1016/j.disc.2024.114350

Klavdija Kutnar , Dragan Marušič , Andriaherimanana Sarobidy Razafimahatratra

Motivated by Gregor et al. (2023) [7], existence of Hamilton cycles, admitting large rotational symmetry, in certain vertex-transitive graphs is investigated. Given a graph X with a Hamilton cycle C, the compression factor

κ (X, C)

of C is the order of the largest cyclic subgroup of

Aut (C) \cap Aut (X)

, and the Hamilton compression

κ (X)

of X is the maximum compression factor over all of its Hamilton cycles. It is shown that for

p, q

distinct primes, vertex-primitive graphs of order pq have Hamilton compression equal to p or q. In addition, for each

n = 1 + 2^{2^{e}}

e > 1

, a connected vertex-transitive graph of order 3n and Hamilton compression equal to n is constructed. As a consequence Hamilton compressions of vertex-transitive graphs of order 3p, p a prime, are determined. Similarly, Hamilton compressions of vertex-transitive graphs of order 2p, p a prime, are also computed.

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引用次数: 0

Spectral extrema of graphs: Forbidden star-path forests

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-10 DOI: 10.1016/j.disc.2024.114351

Yanni Zhai , Xiying Yuan , Lihua You

A path of order n is denoted by

P_{n}

, and a star of order n is denoted by

S_{n - 1}

. Recently, Fang and Yuan determined the Turán numbers of

k S_{ℓ - 1} \cup P_{ℓ}

k_{1} S_{2 ℓ - 1} \cup k_{2} P_{2 ℓ}

and

k S_{4} \cup 2 P_{5}

for n appropriately large. In this paper, we extend the results to the spectral counterpart. The graphs with maximum spectral radii among graphs containing no any copy of these three kinds of star-path forests are completely characterized.

引用次数: 0

Monochromatic cycles in 2-edge-colored bipartite graphs with large minimum degree

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-09 DOI: 10.1016/j.disc.2024.114363

Yiran Zhang , Yuejian Peng

<div><div>For graphs G, <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, we write <math><mi>G</mi><mo>⟼</mo><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> if any red-blue edge coloring of G yields a red <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> or a blue <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. The Ramsey number <math><mi>r</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is the minimum number n such that the complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>⟼</mo><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. There is an interesting phenomenon that for some graphs <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> there is a number <math><mn>0</mn><mo><</mo><mi>c</mi><mo><</mo><mn>1</mn></math> such that for any graph G of order <math><mi>r</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> with minimum degree <math><mi>δ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>></mo><mi>c</mi><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo></math>, <math><mi>G</mi><mo>⟼</mo><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. When we focus on bipartite graphs, the bipartite Ramsey number <math><mi>b</mi><mi>r</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math> is the minimum number n such that the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>⟼</mo><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math>. Previous known related results on cycles are on the diagonal case (<math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math><

引用次数: 0

Some orientation theorems for restricted DP-colorings of graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-09 DOI: 10.1016/j.disc.2024.114352

Ian Gossett

We define Z-signable correspondence assignments on multigraphs, which generalize good correspondence assignments as introduced by Kaul and Mudrock. We introduce an auxiliary digraph that allows us to prove an Alon-Tarsi style theorem for DP-colorings from Z-signable correspondence assignments on multigraphs, and apply this theorem to obtain three DP-coloring analogs of the Alon-Tarsi theorem for arbitrary correspondence assignments as corollaries. We illustrate the use of these corollaries for DP-colorings on a restricted class of correspondence assignments on toroidal grids.

引用次数: 0

On finite groups whose power graph is claw-free

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-05 DOI: 10.1016/j.disc.2024.114348

Pallabi Manna , Santanu Mandal , Andrea Lucchini

Let G be a finite group and let

P (G)

be the undirected power graph of G. Recall that the vertices of

P (G)

are labelled by the elements of G, with an edge between

g_{1}

and

g_{2}

if either

g_{1} \in 〈 g_{2} 〉

g_{2} \in 〈 g_{1} 〉

. The subgraph induced by the non-identity elements is called the reduced power graph, denoted by

P^{⁎} (G)

. The main purpose of this paper is to investigate the finite groups whose reduced power graph is claw-free, which means that it has no vertex with three pairwise non-adjacent neighbours. In particular, we prove that if

P^{⁎} (G)

is claw-free, then either G is solvable or G is an almost simple group. In the second case, the socle of G is isomorphic to

PSL (2, q)

for suitable choices of q. Finally we prove that if

P^{⁎} (G)

is claw-free, then the order of G is divisible by at most 5 different primes.

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引用次数: 0

Unavoidable induced subgraphs of infinite 2-connected graphs

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-04 DOI: 10.1016/j.disc.2024.114346

Sarah Allred , Guoli Ding , Bogdan Oporowski

In 1930, Ramsey proved that every infinite graph contains either an infinite clique or an infinite independent set. König proved that every connected infinite graph contains either a ray or a vertex of infinite degree. In this paper, we establish the 2-connected analog of these results.

引用次数: 0

On 13-crossing-critical graphs with arbitrarily large degrees

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-12-02 DOI: 10.1016/j.disc.2024.114347

Petr Hliněný, Michal Korbela

A recent result of Bokal et al. (2022) [3] proved that the exact minimum value of c such that c-crossing-critical graphs do not have bounded maximum degree is

c = 13

. The key to that result is an inductive construction of a family of 13-crossing-critical graphs with many vertices of arbitrarily high degrees. While the inductive part of the construction is rather easy, it all relies on the fact that a certain 17-vertex base graph has the crossing number 13, which was originally verified only by a machine-readable computer proof. We provide a relatively short self-contained computer-free proof of the latter fact. Furthermore, we subsequently generalize the critical construction in order to provide a definitive answer to another long-standing question of this research area; we prove that for every

c \geq 13

and integers

d, q

, there exists a c-crossing-critical graph with more than q vertices of each of the degrees

3, 4, \dots, d

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引用次数: 0

Regular graphs to induce even periodic Grover walks 正则图来诱导周期格罗弗行走

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-29 DOI: 10.1016/j.disc.2024.114345

Sho Kubota , Hiroto Sekido , Kiyoto Yoshino

The interest of this paper is a characterization of graphs that induce periodic Grover walks with given periods. In previous studies, Yoshie has shown that the only graphs that induce odd periodic Grover walks are cycle graphs. However, this problem is largely unsolved for even periods. In this study, we show that regular graphs that induce 2l-periodic Grover walks are also cycle graphs in most cases, where l is an odd integer. The proof uses Galois theory.

本文的兴趣是对具有给定周期的格罗弗周期行走图的刻画。在之前的研究中，Yoshie已经证明了唯一能引起奇周期格罗弗行走的图是循环图。然而，这个问题在很大程度上没有得到解决。在本研究中，我们证明了在大多数情况下，引起2l周期Grover行走的正则图也是循环图，其中l是一个奇数。证明使用伽罗瓦理论。

引用次数: 0

Bivariate P- and Q-polynomial structures of the association schemes based on attenuated spaces 基于衰减空间的关联方案的二元P-和q -多项式结构

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-28 DOI: 10.1016/j.disc.2024.114332

Pierre-Antoine Bernard , Nicolas Crampé , Luc Vinet , Meri Zaimi , Xiaohong Zhang

The bivariate P- and Q-polynomial structures of association schemes based on attenuated spaces are examined using recurrence and difference relations of the bivariate polynomials which form the eigenvalues of the scheme. These bispectral properties are obtained from contiguity relations of univariate dual q-Hahn and affine q-Krawtchouk polynomials. The bispectral algebra associated to the bivariate polynomials is investigated, as well as the subconstituent algebra of the schemes. The properties of the schemes are compared to those of the non-binary Johnson schemes through a limit.

利用构成组合方案特征值的二元多项式的递推关系和差分关系，研究了基于衰减空间的组合方案的二元P-和q -多项式结构。这些双谱性质是由单变量对偶q-Hahn多项式和仿射q-Krawtchouk多项式的邻接关系得到的。研究了与二元多项式相关的双谱代数，以及方案的子成分代数。通过一个极限，比较了这些格式与非二元Johnson格式的性质。

引用次数: 0

Palindromicity of the numerator of a statistical generating function 统计生成函数分子的重码性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-27 DOI: 10.1016/j.disc.2024.114336

Rebecca Bourn , William Q. Erickson

We prove a conjecture of Bourn and Willenbring (2020) regarding the palindromicity and unimodality of a certain family of polynomials

N_{n} (t)

. These recursively defined polynomials arise as the numerators of generating functions in the context of the discrete one-dimensional earth mover's distance (EMD). The key to our proof is showing that the defining recursion can be viewed as describing sums of symmetric differences of pairs of Young diagrams; in this setting, palindromicity is equivalent to the preservation of the symmetric difference under the transposition of diagrams. We also observe a connection to recent work by Defant et al. (2024) on the Wiener index of minuscule lattices, which we reinterpret combinatorially to obtain explicit formulas for the coefficients of

N_{n} (t)

and for the expected value of the discrete EMD.

我们证明了 Bourn 和 Willenbring（2020 年）关于 Nn(t)多项式族的宫调性和单模性的猜想。这些递归定义的多项式在离散一维地球移动距离（EMD）中作为生成函数的分子出现。我们证明的关键在于证明定义递归可以看作是描述杨图对的对称差之和；在这种情况下，对偶性等同于在图的转置下保持对称差。我们还观察到了与 Defant 等人（2024 年）最近关于微小网格的维纳指数的研究的联系，我们对其进行了组合解释，从而得到了 Nn(t) 的系数和离散 EMD 的期望值的明确公式。

引用次数: 0

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